ChemWiki - User contributions [en]

Rep:Mod:annie91

2013-12-06T16:40:33Z

Aln09: /* Diels Alder Cycloaddition */

=Cope Rearrangement=

Throughout this report click on the hyperlinks to view the log file for the relevant calculation.

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

By looking at the log file it is clear the calculation converges to find a minimum energy, hence confirming the transition state.

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 5: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in Table6.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table 6: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}
====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Conclusion==

Throughout this exercise many different optimisation techniques have been employed, it is clear that the use of one single technique is insufficent. It is much more computationally efficient to optimise at a lower level of theory and continue further optimisations at higher levels of theory. Methods such as IRC can then be used to confirm the geometry of the structures.

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:35:50Z

Aln09: /* IRC Calculation */

=Cope Rearrangement=

Throughout this report click on the hyperlinks to view the log file for the relevant calculation.

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

By looking at the log file it is clear the calculation converges to find a minimum energy, hence confirming the transition state.

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 5: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in Table6.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table 6: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:34:36Z

Aln09: /* Diels Alder Cycloaddition */

=Cope Rearrangement=

Throughout this report click on the hyperlinks to view the log file for the relevant calculation.

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 5: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in Table6.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table 6: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:31:56Z

Aln09: /* Cope Rearrangement */

=Cope Rearrangement=

Throughout this report click on the hyperlinks to view the log file for the relevant calculation.

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 5: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in Table6.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table 6: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:31:00Z

Aln09: /* Diels Alder Cycloaddition */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 5: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in Table6.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table 6: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:29:47Z

Aln09: /* Cis-butadiene */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table 4: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:29:26Z

Aln09: /* Freezing Reaction Coordinate using Redudant Coordinate Editor */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table 3: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:29:07Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 1: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table 2: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:28:39Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

Sum of electronic and zero-point Energies= -234.468193
Sum of electronic and thermal Energies= -234.460941
Sum of electronic and thermal Enthalpies= -234.459997
Sum of electronic and thermal Free Energies= -234.499482

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:27:33Z

Aln09: /* Regioselectivity of Diels Alder */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:25:22Z

Aln09: /* Regioselectivity of Diels Alder */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:23:40Z

Aln09: /* Regioselectivity of Diels Alder */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

Diels Alder Cycloaddition can proceed via an Exo or Endo mechanism:

[[File:Endo mechanism.jpg |400px|left|thumb|Endo Mechanism]]
[[File:Exo mechanism.jpg |400px|right|thumb|Exo Mechanism]]

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

File:Endo mechanism.jpg

2013-12-06T16:16:59Z

Aln09: uploaded a new version of "File:Endo mechanism.jpg"

File:Exo mechanism.jpg

2013-12-06T16:16:31Z

Aln09: uploaded a new version of "File:Exo mechanism.jpg"

Rep:Mod:annie91

2013-12-06T16:12:39Z

Aln09: /* Diels Alder Cycloaddition */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:10:46Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

The anti2 structure was optimised under a HF/3-21G level of theory to give an energy of -231.69097a.u., and was then reoptimised to the higher level of theory B3LYP/6-31(d) to give an energy of -234.61079.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T16:00:17Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:56:24Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:55:31Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:54:40Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:54:26Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:49:55Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

 The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -234.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/6/65/REACT_GAUCHE_DFT.LOG -234.609110]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

File:REACT GAUCHE DFT.LOG

2013-12-06T15:49:12Z

Aln09:

Rep:Mod:annie91

2013-12-06T15:44:35Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

 The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -232.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:43:58Z

Aln09: /* Cope Rearrangement */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}
 

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -232.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:42:12Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using HF/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}
 

The resulting structures were then re-optimised under a B3LYP/6-31(d). (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/5/57/REACT_ANTIDFTOPT.LOG -232.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

 Both optimisations confirmed the antiperiplanar conformation (dihedral angle 180o) had a lower energy than the gauche conformation (dihedral angle 60o). This in agreement with the hypothesis that the due to steric contraints the gauche conformation will have a greater energy.

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

File:REACT ANTIDFTOPT.LOG

2013-12-06T15:37:33Z

Aln09:

Rep:Mod:annie91

2013-12-06T15:35:05Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/3-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: B3LYP/6-31(d) Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -232.5597]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:29:39Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:left;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|left|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:28:43Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:27:53Z

Aln09: /* Energies and MO Analysis */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:20:47Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T15:19:22Z

Aln09: /* Energies and MO Analysis */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: HF/£-21G Optimisation of 1,5-Hexadiene Confomers
! Confomer !! Energy (a.u.) !! Point Group
|- align="center"
! scope="row"| Anti
| [https://wiki.ch.ic.ac.uk/wiki/images/a/ad/REACT_ANTI.LOG -231.69254]
| Ci
|- align="center"
! scope="row"| Gauche
| [https://wiki.ch.ic.ac.uk/wiki/images/8/80/REACT_GAUCHE.LOG -231.68961]
| C1
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

File:REACT GAUCHE.LOG

2013-12-06T15:17:25Z

Aln09: uploaded a new version of "File:REACT GAUCHE.LOG"

File:REACT ANTI.LOG

2013-12-06T15:17:24Z

Aln09: uploaded a new version of "File:REACT ANTI.LOG"

Rep:Mod:annie91

2013-12-06T15:12:40Z

Aln09: /* Optimisation of Reactants and Products */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

The following anti periplanar and gauche confomers of 1,5-hexadiene was optimised using Hf/3-21g level of theory, their energies are tabulated below. (See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.)

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T14:42:11Z

Aln09: /* Further B3LYP/6-31(d) Optimisation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/b7/CHAIR_TS_GUESSOPTD.LOG -231.619322]
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/2/2e/CHAIR_TS_OPTD_B3LYP2.LOG -234.556982]
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/0/0d/BOAT_QST2BOATLIKE.LOG -231.602802]
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:CHAIR_TS_OPTD_B3LYP2.LOG -234.543093]
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/c/c1/REACT_ANTI2_3.LOG -231.690970]
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | [https://wiki.ch.ic.ac.uk/wiki/images/b/ba/REACT_ANTI2_3DFTOPT.LOG -234.610789 ]
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

File:REACT ANTI2 3DFTOPT.LOG

2013-12-06T14:34:23Z

Aln09:

File:REACT ANTI2 3.LOG

2013-12-06T14:34:22Z

Aln09:

File:CHAIR TS GUESSOPTD.LOG

2013-12-06T14:34:21Z

Aln09:

File:BOAT QST2BOATLIKE.LOG

2013-12-06T14:34:21Z

Aln09:

File:CHAIR TS OPTD B3LYP2.LOG

2013-12-06T14:34:20Z

Aln09:

File:BOAT QST2BOATLIKE B3LYP.LOG

2013-12-06T14:34:19Z

Aln09:

Rep:Mod:annie91

2013-12-06T14:31:43Z

Aln09: /* Further B3LYP/6-31(d) Optimisation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556982
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.690970
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | -234.610789
| align="center" | -234.468193
| align="center" | -234.460941
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.7
| align="center" | 43.7
| align="center" | 33.4
| align="center" | 32.6
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 54.5
| align="center" | 53.8
| align="center" | 41.3
| align="center" | 40.7
| align="center" | 44.7 ± 2.0
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T14:30:20Z

Aln09: /* Further B3LYP/6-31(d) Optimisation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

The boat and chair structures undergo further optimisation under the B3LYP/6-31(d), and the activation energies calculated. The results are shown in the table below. The differences in the energies are significantly different, however the geometries appear to be very similar. Hence it can be concluded that it most efficient to first optimise a given structure at a lower level of the theory, and then reoptimise at a higher level.

(The format of the table was taken from the script)

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461343
| align="center" | -234.556982
| align="center" | -234.414934
| align="center" | -234.409011
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450930
| align="center" | -231.445301
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.690970
| align="center" | -231.537850
| align="center" | -231.530962
| align="center" | -234.610789
| align="center" | -234.468193
| align="center" | -234.460941
|}

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T14:21:41Z

Aln09: /* IRC Calculation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

Item Value Threshold Converged?
Maximum Force 0.000009 0.000450 YES
RMS Force 0.000003 0.000300 YES
Maximum Displacement 0.000092 0.001800 YES
RMS Displacement 0.000035 0.001200 YES
Predicted change in Energy=-1.838680D-09
Optimization completed.
-- Stationary point found.

===Further B3LYP/6-31(d) Optimisation===

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T13:22:44Z

Aln09: /* Cope Rearrangement */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

===Further B3LYP/6-31(d) Optimisation===

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T13:21:35Z

Aln09: /* IRC Calculation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

===Further B3LYP/6-31(d) Optimisation===

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T13:04:50Z

Aln09:

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>

Rep:Mod:annie91

2013-12-06T13:04:22Z

Aln09: /* IRC Calculation */

=Cope Rearrangement=

==Optimisation of Reactants and Products==

===Anti===

The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory. See [[Mod:annie91#Appendix_1|Appendix 1]] for summary table.

[[File:Anti nofer.jpg|250px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]

===Gauche===
The following anti periplanar confomer of 1,5-hexadiene was optimised using Hf/3-21g level of theory.

[[File:Gauche nofer.jpg|250px|right|thumb|HF Optimised Gauche Conformation of 1,5-hexadiene.]]

==Optimisation of Transition Structures==

The boat and chair transition states were optimised using a variety of the methods to find the most stable geometry, summary tables of all the calculation can be found in [[Mod:annie91#Appendix_1|Appendix 1]].

===Chair===

====Calculation of Force Constant Matrix====

A "guess" structure from two allyl (C3H5)was optimised to a TS(Berny) using '''HF/3-21G''' level of theory. The terminal carbons were positioned in an eclipsed fashion approximately 2.2Å apart. When using this method of optimisation is particularu important to guess structure is close to the true geometry ensure a successful calculation.

====Freezing Reaction Coordinate using '''Redudant Coordinate Editor'''====
[[File:Chair modredopt.jpg|200px|right|thumb|HF Optimised Anti Conformation of 1,5-hexadiene.]]
 The terminal carbons of the "guess structure" were frozen at approximately 2.2Å apart, and the structure was optimised to a minimum using the'''HF/3-21G''' level of theory. The bonds were then unfrozen and optimised to a TS(Berny) using '''HF/3-21G''' level of theory.
{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|+ Table XX: Chair Tranistion States Data
|- align="center"
! scope="col"|
! scope="col"| Hessian
! scope="col"| ModRedundant
! scope="col"| Combination
|- align="center"
! scope="row"| Energy (a.u.)
| -231.619322
| -231.615979
| -231.619322
|- align="center"
! scope="row"| Frequency cm-1
| -817.76
| -
| -817.81
|- align="center"
! scope="row"| Bond Length Å '''C(1)-C(9)'''
| 2.02
| 2.17
| 2.02
|- align="center"
! scope="row"| Bond Length Å '''C(6)- C(14)'''
| 2.02
| 2.2
| 2.02
|}

Calculation of the hessian and differentiating along the reaction coordinate yeilded similar results, indicating. that the guess structure was close to the true structure. Both techniques are reliable, however as the Modredundant technique is often preferred when the guess structure is more complex.

===Boat===

The boat geometry was optimised using the QTS2 method. This technique requires the structures of the reactant and produts (both numbered in the same fashion) and interpolates between them to find the transition state. The anti conformation was numbered in the following way and the TS(QTS2) calculation performed.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) First Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Fail input structures boat.jpg|600px|centre|thumb|]]
| [[File:Boat fail video.gif|300px|centre|thumb|]]
|}

As it can clearly be seen the calculation failed, and the transition structure produced was very similar to the chair geometry. This is because the guess structure was too disimilar from the transition state. A second minimisation was then run after manual modification of the reactant and product to resemble the boat geometry more closely.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XX: TS(QTS2) Second Optimisation
|- align="center"
! scope="col"| Input Geometry
! scope="col"| Output Geometry
|- align="center"
| [[File:Correct input structures boat.jpg|400px|centre|thumb|]]
| [[File:Boat TS QST2 Optimised2.gif|400px|centre|thumb|]]
|}

The resulting tranistion state is the boat structure, hence the manipulation of the reactants and products was necessary.

===IRC Calculation===

IRC (Intrinsic Reaction Coordinate) calculations are necessary to predict the conformation of the 1,5-hexadiene formed, as the geometry of the transition state is not sufficient. The IRC follows the minimum energy from a given transition state to a product. The optimised chair transition state was further optimised using this technique, the force constant energy was calculated every 10 points up to a maximum of 50. However the calculation stopped at 43 indicating a minimum energy structure had already been obtained.

[[File:Irc graph.jpg|1000px]]

The final point on the IRC was reoptimised to a minimum under the HF/3-21G level of theory, decreasing the energy from -231.691530a.u. to -231.691667a.u., confirming the product resulting from the chair tranisition state is the gauche2 conformation.

[[File:Strucuture 43.jpg|200px|right|thumb|Reoptimised Structure 43, geometry is identical to the IRC method]]

=Diels Alder Cycloaddition=

===Cis-butadiene===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Cis-Butadiene MO Anaylsis
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.227
| -0.030
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Antisymmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Cisbutadiene Bonding MOs.jpg|200px|centre|thumb|Cis-Butadiene HOMO.]]
| [[File:Cisbutadiene Antibonding MOs.jpg|200px|centre|thumb|Cis-Butadiene LUMO.]]
|}

 The structure of cis butadiene was optimised using an Semi-Empirical/AM1 level of theory, resulting in an overall energy of [https://wiki.ch.ic.ac.uk/wiki/images/2/26/CIS_BUTADIENE_OPT_AM1.LOG 0.0487972a.u.]. This structure was re-optimised using B3LYP/6-31G(d) level of theory, resulting in a significantly lower energy of [https://wiki.ch.ic.ac.uk/wiki/images/8/8d/CIS_BUTADIENE_OPT_B3YLP.LOG -155.985950a.u] (summary tables can viewed in [[Mod:annie91#Appendix_2|Appendix 2]]). The B3LYP/6-31G(d)generated the lowest energy hence is the more reliable optimisation technique, and will be used for the following calculations.

===Cyclohexene===

====Energies and MO Analysis====

{| class="wikitable" border="1" style="float:right;" "margin: 1em auto 1em auto;"
|+ Table XX: Energies of Cyclohexene Transition State
! Optimisation Technique !! Energy (a.u.)
|- align="center"
! scope="row"| B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:DIELS_ALDER_TS_OPT_HOME.LOG -234.543896'''96''']
|- align="center"
! scope="row"| Mod Redundant and B3LYP/6-31(d)
| [https://wiki.ch.ic.ac.uk/wiki/images/8/8c/DIELS_ALDER_TS_OPTDFT2.LOG -234.543896'''48''']
|}

 The cyclohexene transition state was optimised directly from the 'guess' structure, and then again using the ModRedundant technique (the newly forming σ bonds frozen at approximately 2.2Å), the results are shown in table XX. The mod redundant technique resulted in a slightly lower energy, and hence more stable transition state, this structure undergoes further MO analysis shown in TableXXX.

{| class="wikitable" style="float:center;" "margin: 1em auto 1em auto;"
|+ Table XXX: Molecular Orbitals of the Cyclohexene Transition State
|- align="center"
! scope="col"|
! scope="col"| HOMO
! scope="col"| LUMO
|- align="center"
! scope="row"| Relative Energy (a.u.)
| -0.219
| -0.009
|- align="center"
! scope="row"| Classification
| Bonding
| Antibonding
|- align="center"
! scope="row"| Symmetry about Vertical Plane
| Symmetric
| Symmetric
|- align="center"
! scope="row"| Symmetry about Horizontal Plane
| Antisymmetric
| Antisymmetric
|- align="center"
! scope="row"| Diagram
| [[File:Diels alder ts HOMO.jpg|250px|left|thumb|HOMO of Cyclohexene Transition State.]]
| [[File:Cyclohexene Antibonding MOs.jpg|250px|right|thumb|LUMO of Cyclohexene Transition State.]]
|}

====Bond Lengths Analysis====

{| class="wikitable" style="float:right;"
! Geometry of Cyclohexene Transition State
|- align="center"
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>300</size><script>spin 15;measure 11 2; measure 12 7; measure 2 1; measure 7 6; measure 11 12;</script>
<uploadedFileContents>Diels_alder_ts_image2.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

 The standard C-C bond length between two sp3 hybridised Carbon atoms is 1.54Å<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The Van der Waals radius of a carbon atom is 1.70Å<ref name="VDW">A. Bondi, ''J. Phys. Chem.'', 1964, '''68(3)'''</ref>, hence when two carbon atoms are 3.4Å apart this distance is the closest two atoms can get colliding without there being bonding interaction. The distance between the two terminal Carbon atoms is 2.27Å, this is significantly shorter than the sum of the Van der Waals radii, however is not recognised as a formal bond, hence it can be said there is bonding interaction, indicating the formation of the two new σ bonds.
The standard C=C bond length between two sp2 hybridised Carbon atoms is 1.34Å
<ref name="Bond Legnths">I. Alborta, I. Rozas and J. Elguero, ''Structural Chemistry'', 1998, '''115(4)'''</ref>. The length of the C=C bonds in the tranisition state are 1.38Å and 1.39Å, this lengthening is in accordance with the formation of a C-C bond between two sp3 hybridised Carbon atoms.

====Vibration Analysis====

The computed imaginary frequency exhibits the molecule vibrating in a synchronous motion, which confirms the predicted concerted syn cylcoaddition mechanism. In contrast the lowest positive frequency demonstrates the translational asynchronous motion.

{| class="wikitable" style="float:centre;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-525.23cm-1 || Lowest Positive Frequency = 135.73cm-1
|- align="center"
|[[File:Cyclohexen TS movie.gif|300px|centre|thumb|]]|| [[File:Cyclohexen TS movie lowest positive freq.gif|300px|centre|thumb|]]
|}

===Regioselectivity of Diels Alder===

{| class="wikitable" style="float:right;" "margin: 1em auto 1em auto;"
|- align="center"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| Structure
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20;measure 22 11; measure 20 7; measure 2 16; measure 5 15;</script>
<uploadedFileContents>ENDO TS OPT DFT MODRED2 CHKTRIAL.mol</uploadedFileContents>
</jmolApplet></jmol>
| <jmol><jmolApplet><title>Geometry of Cyclohexene Transition State</title><color>lightgrey</color>
<size>250</size><script>spin 20; measure 22 14; measure 20 9; measure 2 16; measure 5 15 </script>
<uploadedFileContents>Exo ts jmol.mol</uploadedFileContents>
</jmolApplet></jmol>
|- align="center"
! scope="row"| Energy (a.u.)
| [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.]
| [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.]
|- align="center"
! scope="row"| Distance Between O=C-O-C=O fragment and "Opposite" C/Å
| 3.14
| 2.53
|- align="center"
! scope="row"| Partially Formed C-C σ Bond/Å
| 2.27
| 2.29
|}

 Using B3LYP/6-31(d) optimisation the endo transition had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/f/ff/ENDO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.683397a.u.] , and the exo transition state had a total energy of [https://wiki.ch.ic.ac.uk/wiki/images/9/9e/EXO_TS_OPT_DFT_MODRED2_CHKTRIAL.LOG -612.679311a.u.] .

 The endo adduct is lower in energy than the exo adduct; and as the endo product has been shown to be the major product this confirms that the reaction is under kinetic control.

====Steric Analysis====

Through analysis of the bond lengths it is clear the exo transition state is subject to significantly greater steric strain, accounting for its relative instability. The distance between O=C-O-C=O and the 'opposite' Hydrogen of the CH2-CH2 moiety of the diene is 0.61Å closer than the equivalent distance to the CH-CH moiety of the endo adduct. This strain is particularly large, as this C-H distance in the exo transition state is less than the sum of the Van Der Waals radius (2.79Å<ref>R. Scott Rowland and R. Taylor, ''J. Phys. Chem'', 1996, '''100(18)''', 7384-7391</ref> resulting in orbital replusion. As a result the C=C of the dienophile is not as proximal, hence the favourable HOMO-LUMO overlap is reduced and the partially formed C-C σ bonds are longer, therefore weaker giving rise to a less stable exo transition state. This structure is a compromise between the optimising the favourable HOMO-LUMO interactions whilst minimising the steric repulsion of the CH2-CH2 and O=C-O-C=O fragments.

====Vibration====

The table below shows the vibrational animation of both the endo and exo transition states. Each transition states gives rise to one imaginary frequency, confirming the transition state.

{| class="wikitable" style="float:center;"
! colspan="2"| Vibrations of Cyclohexene Transition State
|- align="center"
|Imaginary Frequency =-446.17cm-1 || Lowest Positive Frequency = 448.49cm-1
|- align="center"
|[[File:Endo ts movie.gif|300px|centre|thumb|]]|| [[File:Exo ts movie.gif|300px|centre|thumb|]]
|}

====Molecular Orbital Analysis====

Similarly to the cis-butadiene and ethylene example, the HOMO of the tranistion states are a result of the diene contributing its HOMO, and the dienophile contributing its LUMO to achieve the most effective orbital overlap. Secondary orbital overlap is accountable for the increased relative stability of the endo transition state. As can be seen in the LUMO+1 orbital of the endo adduct, π electrons from the newly forming C=C bond in the diene are donated in the π* C=O orbital of the maleic anhydride. This is possible due to correct symmetry and similar energies, resulting in effective orbital overlap between the π and π* orbitals. The resulting endo transition state is therefore stabilised. In the case of the exo adduct, such orbital overlap is impossible as the C=O π* are on the opposite side of the molecule, hence the exo adduct does not benefit from any secondary orbital stabilisation. This secondary orbital overlap also accounts for the shorter partially formed C-C bond.

By examination of the HOMOs, it is clear the endo structure has a higher electron density at the O=C-O-C=O moiety,

{| class="wikitable" style="margin: 1em auto 1em auto;"
! scope="col"|
! scope="col"| Endo
! scope="col"| Exo
|- align="center"
! scope="row"| HOMO
| [[File:Endo HOMOS.jpg|450px|]] Relative Energy = -0.24223a.u. Antisymmetric
| [[File:Exo HOMOS.jpg|450px|]] Relative Energy = - 0.24216a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO
| [[File:Endo LUMOS.jpg|450px|]] Relative Energy = -0.06782a.u. Antisymmetric
| [[File:Exo LUMOs.jpg|450px|]] Relative Energy = - 0.07840a.u. Antisymmetric
|- align="center"
! scope="row"| LUMO+1
| [[File:Endo LUMOS+1.jpg|450px|]] Relative Energy = -0.05265a.u. Symmetric
| [[File:Exo LUMOS+1.jpg|450px|]] Relative Energy = - 0.05188a.u. Symmetric
|}

==Appendix 1==

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6925353a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001891a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Gauche Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6896157a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00001053a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0.4439 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Anti2 Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -234.55970423a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.0000135a.u.
|-
! scope="row"| Imaginary Frequency
|
|-
! scope="row"| Dipole Moment
| 0 Debye
|-
! scope="row"| Point Group
| Ci
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Hessian Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.6193223a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003503a.u.
|-
! scope="row"| Imaginary Frequency
| -817.76cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair ModRedundant Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61597903a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00333031a.u.
|-
! scope="row"| Imaginary Frequency
| -
|-
! scope="row"| Dipole Moment
| 0.0073 Debye
|-
! scope="row"| Point Group
| C1
|}

{| class="wikitable" style="float:left;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Chair Combination Optimisation
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FREQ
|-
! scope="row"| Calculation Method
| RHF
|-
! scope="row"| Basis Set
| 3-21G
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -231.61932232a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00003475a.u.
|-
! scope="row"| Imaginary Frequency
| -817.81cm-1</sup.
|-
! scope="row"| Dipole Moment
| 0.0009 Debye
|-
! scope="row"| Point Group
| C1
|}

==Appendix 2==

{| class="wikitable" style="float:centre;" "margin: 1em auto 1em auto;"
|-
!colspan="2"|Cis-Butadiene Summary
|-
! scope="row"| File Type
| .chk
|-
! scope="row"| Calculation Type
| FOPT
|-
! scope="row"| Calculation Method
| RB3LYP
|-
! scope="row"| Basis Set
| 6-31G(D)
|-
! scope="row"| Charge
| 0
|-
! scope="row"| Spin
| Singlet
|-
! scope="row"| Total Energy
| -155.98594952a.u.
|-
! scope="row"| RMS Gradient Norm
| 0.00005672a.u.
|-
! scope="row"| Dipole Moment
| 0.085 Debye
|-
! scope="row"| Point Group
| C2v
|}

==References==
<references/>