=Transition States and Reactivity=
The transition state is a first-order saddle point on the potential energy surface, where it corresponds to a minimum at all points except one. The Hammond–Leffler postulate states that the transition state resembles either reactants or products. Methods used below are based on this postulate.The geometries of the transition states are studied computationally using Gaussview.Different calculation methods are introduced and compared. In this exercise, we cover two types of reactions: the Cope Rearrangement and the Diels-Alder reaction.

==The Cope Rearrangement==
The allowed antiperiplanar and gauche conformations of 1,5-hexadiene are optimized to find the energy-minima and symmetrized to find the point group. The potential energies are calculated using energy optimization to a minimum, the thermochemical data are calculated from a frequency optimization. Usually a combination of OPt+Fre is used.
[[File:Cope REARRANGEMENT.jpg|frame|center|The Cope Rearrangement]]

===Optimization of Reactants and Products===

The conformers of 1,5-hexadiene were optimized to minimum using HF/3-21G method. For each conformer, frequency analysis gives all positive vibrations which indicates it reaches the minimum. The table below shows the conformers and their relative energies. The Gauche 3 conformer is the global minimum because of the attractive interaction between the π orbital and the vinyl protons. However, the energy difference between the gauche3 and anti2 conformations is negligible.<ref name="xxx"/>

{| class="wikitable"
|-
! Conformer !! Structure !! Energy/Hartrees !! Relative Energy/kcal/mol !! Point Group
|-
| Gauche 1|| <jmol><jmolApplet>
<title>Gauche 1</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 1 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68772 || 3.10 || C2
|-
| Gauche 2|| <jmol><jmolApplet>
<title>Gauche 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche2 .mol</uploadedFileContents>
</jmolApplet></jmol> || -231.69167 || 0.62 || C2
|-
| Gauche 3|| <jmol><jmolApplet>
<title>Gauche 3</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 3 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69266 || 0.00 || C1
|-
| Gauche 4||<jmol><jmolApplet>
<title>Gauche 4</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche4 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69153 || 0.71 || C2
|-
| Gauche 5|| <jmol><jmolApplet>
<title>Gauche 5</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE5 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68962 || 1.91 || C1
|-
| Gauche 6|| <jmol><jmolApplet>
<title>Gauche 6</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE6 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68916 || 2.20|| C1
|-
| Anti 2|| <jmol><jmolApplet>
<title>Anti 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 2 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69254 || 0.08 || Ci
|-
| Anti 3|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTI_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68907 || 2.25 || C2h
|-
| Anti 4|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 4 .mol</uploadedFileContents>
</jmolApplet></jmol>||-231.69097 ||1.06 ||C1
|}

The Anti 2 conformer was reoptimized using B3LYP/6-31G(d) method.This method gives a lower energy Anti 2 conformer. 6-31G level gives a higher accuracy than the basis set 3-21G.

HF/3-21G[[File:C6H10_ANTICi_OPT.LOG]]|;
B3LYP/6-31G(d)[[File:ANTI CI 6-31G(d).LOG]]

{| class="wikitable" border="1"
|+ Comparison of different optimisation methods
! Method!! Energy/Hartrees!!Relative energy/kcal/mol !!Convergence !!Jmol
|-
| HF/3-21G|| -231.69254|| 0.08|| Item Value Threshold Converged?
Maximum Force 0.000042 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001524 0.001800 YES
RMS Displacement 0.000749 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| B3LYP/6-31G(d)|| -234.61171 || ||Item Value Threshold Converged?
Maximum Force 0.000049 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001410 0.001800 YES
RMS Displacement 0.000469 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_DFT_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

The table below shows the comparison in bond lengths and bond angles using two methods.

{| class="wikitable" border="1"
|+ Geometry comparison of different optimisation methods
! Parameters!! HF-3-21G!!B3LYP/6-31G!!Literature
|-
| C1=C4|| 1.3162|| 1.3382|| 1.3412
|-
| C4-C6|| 1.5089|| 1.5074|| 1.5077
|-
| C6-C8|| 1.5530|| 1.5549|| 1.5362
|-
| C-H(average)||1.0750||1.0997||1.1077
|-
| C1=C4-C6||124.8||122.0||122.5
|-
| C4-C6-C8||111.4||112.6||111.0
|-
| H-C1=C4||121.9||122.0||120.4
|-
| H-C4-C6||115.5||115.5||118.4
|-
| H-C6-H||107.7||106.7||107.1
|-
| C4-C6-C8-C10(Dihedral angle)||179.99||179.97||179.3
|}
Distances in Å, angles in degrees.

A frequency calculation gives all real and positive vibrational frequencies, which indicates it reaches an energy-minimum. Table below gives the comparison of thermochemistry data using two different optimization methods.

{| class="wikitable" border="1"
|+ Thermochemistry data comparison of different optimisation methods
! Type of energies!! Experimental value(B3LYP/6-31G(d))/Hartrees!!Experimental value(HF/3-21G)/Hartrees
|-
| Sum of electronic and zero-point Energies|| -234.469202||-231.539540
|-
| Sum of electronic and thermal Energies|| -234.461856||-231.532566
|-
| Sum of electronic and thermal Enthalpies||-234.460912||-231.531622
|-
| Sum of electronic and thermal Free Energies|| -234.500773||-231.570913
|-
|}

===Optimisation of the Chair and Boat Transition States===

====Chair Transition State====
In the chair transition structure two optimized C3H5 allyl fragments are positioned approximately 2.2 Å apart, with C2h symmetry. Two different methods are set up to optimize the transition structure: Hartree-Fock(3-21G) where the force constants are computed at the beginning of the calculation and the Redundant Coordinate Editor.

=====Hartree-Fock(3-21G) Method=====

[[File:CHAIR TS GUESS ANIMATION.gif|frame|center|Optimized Chair Transition State Animation using Hartree-Fock Method]]

In this Hartree-Fock method, a Gaussian optimization was set up as Opt+Fre with Optimization to a TS(Berny). The force constant was calculated once. This stops the calculation crashing if more than one imaginary frequency is detected.The calculation gives an imaginary frequency of -818 cm-1 and a energy of -231.61932 Hartrees.

This method is only used when you have a resonable guess for the transition structure. The optimization is repeated using DFT/B3YLP/6-31G method, which gives an electronic energy of -234.55698 Hartrees and an imaginary frequency of -566 cm-1.

This is the one corresponding to the Cope rearrangement.

=====Frozen Coordinate Method=====
This method is used when the guessed structure is far away from the transition structure.

The Redundant Coordinate Editor is used to freeze the coordinates. The distance between two of the terminal carbons of the allyl fragments is fixed to be 2.2 Å. The distance between the opposite two of the terminal carbons of the allyl fragments is also fixed to be 2.2Å.

The optimized chair transition structure is like the sturcture optimized using HF/3-21G method, with an electronic energy of -231.69167 Hartrees and a imaginary frequency of -765 cm-1

The two coordinates we differentiate along the path is shown using numerical normal guess Hessian. The newly formed σ C-C bond distance is 1.5509 Å. Compared with the one optimized above using HF/3-21G, this method gives a longer bond forming length.

[[File:CHAIR FROZEN COORDINATE HESSIAN.gif|frame|center|Optimized Chair Transition State Animation using Frozen Coordinate Method with Modified Heissan.]]

====Boat Transition State====
In the boat transition structure two allyl fragments are positioned 2.2Å apart, with C2v symmetry. The structure is optimized using QST2 method, where the calculation will interpolate between reactants and products and locate a transition state in between them. This method would fail if the reactant and product structures are not close enough to the transition state structure.

[[File:BOAT LABEL ARRANGEMENT.PNG|thumb|center|Reactant and Product with Labelled Atoms |500px]]

The structure above is like a more dissociated version of the chair transition structure. The limitation of QST2 method is that the job is only successful when the input file includes a structure closer to the boat transition structure. In order to optimize the transition state the structure is modified as shown below.

[[File:BOAT ARRANGEMENT2.PNG|thumb|center|Reactant and Product with Labelled Atoms and Modified Arrangements|600px]]

QST3 method is more reliable than QST2 because it allows you to input the guessed transition structure.

=====Intrinsic Reaction Coordinate=====

IRC method calculates the minimum energy path from a transition state structure to the local minimum. Since the reaction is symmetrical, the reaction coordinate is computed only in the forward direction. If given an unsymmetrical reaction then the reaction coordinate will be computed in both directions. As the frequency calculation was previously computed, we calculate the force constants once, rather than at every step along the path. This IRC method gives an electronic energy of -231.69153 Hartrees. The animation is shown below.

[[File:CHAIR IRC VIBRATION.gif|frame|center|Optimized Chair Transition State using IRC method]]

The IRC method didn't give a minimum geometry. It only gives the direction where the slope of the potential energy surface is steepest. The chair transition structure is re-optimized using three approaches listed below.

(1)Minimization for the last point on the IRC
Minimization of the last point on the IRC gives an electronic energy of -231.69167 Hartrees. This approach requires the structure close to the chair transition state, otherwise it will give a wrong minimum.

(2)Change the number of points along the IRC to a higher number until it reaches a minimum.From the table, we can tell that an increase in number of points leads to a alsightly lower energy.

{| class="wikitable" border="1"
|+ Optimization along the IRC
! Number of Points!! Electronic Energy
|-
| 100 || -231.69158 Hartrees
|-
| 150 || -231.69158 Hartrees
|-
| 200 || -231.69158 Hartrees
|}

====Summary of Results====

The table below shows the comparison of activation energies in HF/3-21G and B3LYP/6-31G.
{| class="wikitable" border="1"
|+ Summary of Activation Energies(in kcal/mol)
! Terms!!HF/3-21G!! HF/3-21G!!B3LYP/6-31G!!B3LYP/6-31G!!Experimental Value
|-
! !!at 0K!!at 298.15K!! at 0K!!at 298.15K!!at 0K
|-
| ΔE(chair)|| 45.71||44.70||34.07||33.16||33.5±0.5
|-
| ΔE(boat)|| 55.60||54.76||41.96||41.32||44.7±2.0
|}

The table below shows the thermochemical data for chair and boat transition structures.
{| class="wikitable" border="1"
|+ Summary of Energies(in Hartreers)
! !!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
! !!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies!!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies
|-
! !! !! at 0K!! at 298.15K!!!!at 0K!! at 298.15K
|-
| Chair TS|| -231.61932||-231.46670 || -231.46134||-234.55698 || -234.41491 ||-234.40898
|-
| Boat TS|| -231.60280||-231.45093 ||-231.44530|| -234.54308||-234.40236|| -234.39601
|}

==The Diels Alder Cycloaddition==

Diels-Alder is a [4+2] cycloaddition reaction. Diels-Alder reactions occur between a conjugated s-cis diene and a dienophile. In this exercise, the reaction between maleic anhydride and cyclopentadiene is studied. The less stable product, the endo product is formed in the irreversible Diels-Alder reactions. The kinetic product is formed faster because the bonding interaction in transition state between the electron-deficient C=O groups and back of the diene lowers the energy of its transition state.

The HOMO and LUMO interact when there is a significant overlap between them.They must have the same symmetry for a good overlap. In the cis-butadiene and ethylene cycloaddition, the HOMO of ethylene and the LUMO of cis-butadiene are both symmetric with respect to the plane, the LUMO of ethylene and the HOMO of cis-butadiene are both antisymmetric with respect to the plane.

===Optimization of Cis-Butadiene and Ethylene===
The table below shows the HOMO and LUMO of cis-butadiene and ethylene.The structures are optimized using Hartree-Fock/3-21G method and the MOs are visualized with an isovalue of 0.02.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Transition Structures
!colspan="3"|'''Cis-Butadiene'''
!colspan="3"|'''Ethylene'''
|-
! Term!!Molecular Orbitals!! Symmetry!!Term!!Molecular Orbitals!! Symmetry
|-
| LUMO|| [[File:LUMO of cis-butadiene.PNG]] || Symmetric||LUMO|| [[File:LUMO of ETHYLENE.PNG]]|| Anti Symmetric
|-
| HOMO|| [[File:HOMO of CIS BUTADIENE .PNG]]|| Anti symmetric||HOMO|| [[File:HOMO of ETHYLENE.PNG]] || Symmetric
|}

===Computation of the Transition State Geometry===

[[File:PROTOTYPE TS OPT.gif]]

The Hartree-Fock/3-21G method gives an imaginary frequency of -818 cm-1. AM1 Semi-emipirical method gives an imaginary frequency of -956 cm-1. The presence of an imaginary frequency indicates a negative secondary derivative, which corresponds to the transition state. The animation above corresponds to the reaction path,we can tell from it that the formation of the two bonds is synchronous. From the MOs of the transition states we can tell LUMO of ethylene and HOMO of cis-butadiene interact to give an antisymmetric HOMO.HOMO of ethylene and LUMO of cis-butadiene interact to give a symmetric HOMO.

HOMO and LUMO of the transition states are shown below.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Prototype Reaction Transition Structure
!colspan="4"|'''HF/3-21G'''
!colspan="4"|'''AM1 Semi-Empirical'''
|-
! Term!! Molecular Orbitals!! Symmetry!!Relative Energy!!Term!! Molecular Orbitals!! Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of Prototype REACTION TS.PNG]]||Symmetric||0.14241||LUMO||[[File:LUMO of PROTOTYPE TS OPT AM1.PNG]]||Symmetric||0.02315
|-
| HOMO|| [[File:HOMO of Prototyre Reaction TS.PNG]]|| Symmetric||-0.30087||HOMO||[[File:HOMO of PROTOTYPE TS OPT AM1 .PNG]] ||AntiSymmetric||-0.32394
|}

MOs are visualized with an isovalue of 0.02 in both methods.

A typical sp3 C-C bond distance is 1.54 cm. A typical sp2 C-C bond distance is 1.50 cm.
The van der Waals radius of the C atom vibration is 1.7 Å.
The bond length of the partly formed C-C bond is 2.20 Å which is longer than the normal sp3 C-C bond distance and the van der Waals radius.

===Regioselectivity of the Diels Alder Reaction===
The kinetically controlled reaction between cyclohexa-1,3-diene and maleic anhydride is studied by guessing and optimizing its transition state using the method we introduced above. The corresponding energy for each conformation is calculated using Gaussview. The major product formed is the endo product. The endo rule is explained computationally.

[[File:Diels alder 2.png|thumb|center|Diels Alder Reaction between Cyclohexa-1,3-diene and Maleic Anhydride|500px]]

====HOMO and LUMO of Transition States====

The transition structure is optimized using AM1 Semi-empirical method. The reaction path is shown in the animation below.

{| class="wikitable" border="1"
|+ ENDO and EXO Transition States
! Term!! ENDO!!EXO
|-
| Transition Structure|| [[Image:ENDO TS OPT.gif]] || [[Image:EXO OPT TS.gif]]
|-
| Energy|| -0.051505 Hartrees|| -0.050419 Hartrees
|-
| Imaginary Frequency|| -806 cm-1|| -812 cm-1
|}

The less stable product, the endo product is formed in this irreversible Diels-Alder reactions. Besides the usual primary interaction, there is an additional bonding interaction between the electron-deficient C=O grouphes and back of the diene.This interaction is known as a secondary orbital interaction, which lowers the transition state energy relative to the exo product.<ref name="secondary orbital"/>

[[File:SECONDARY ORBITAL.png|thumb|center|Secondary Orbital Interaction|500px]]

{| class="wikitable" border="1"
|+ HOMO and LUMO of ENDO and EXO Transition States
!colspan="4"|'''ENDO'''
!colspan="4"|'''EXO'''
|-
! Term!! Molecular Orbitals!!Symmetry!!Relative Energy!!Term!! Molecular Orbitals!!Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of ENDO.PNG ]]||Antisymmetric || -0.03570 || LUMO || [[File:EXO LUMO .PNG ]] || Antisymmetric || 0.00601
|-
| HOMO|| [[File:HOMO of ENDO.PNG ]]|| Antisymmetric || -0.34505 || HOMO || [[File:EXO HOMO .PNG ]] ||Symmetric || -0.38787
|}

====Intrinsic Reaction Coordinate Calculation====

{| class="wikitable" border="1"
|+ IRC and RMS Gradient of ENDO and EXO Conformations
! Term!! ENDO!!EXO
|-
| IRC path|| [[File:ENDO IRC .PNG]] || [[File:EXO IRC .PNG]]
|-
| RMS Gradient||[[File:ENDO RMS.PNG]] || [[File:EXO RMS.PNG]]
|-
| Relative Energy of Transition State/Hartrees|| -0.051505 ||-0.050420
|-
|Relative Energy of Product at IRC=-5.9/Hartrees||-0.15985||-0.15991
|}

AM1 method gives an IRC path which resembles the path going from product(on the left) to reactants(on the right). The reactants tend to be far away from each other therefore the calculation does not converge.A weird point on the IRC curve is observed after the transition state, where it suddenly drops to the product. However, this point does not show up in the corresponding RMS gradient curve.
IRC curve proves the optimized transition structures for endo and exo products are correct. A correct optimized transition structure has a gradient of 0.

===Further Discussion===
In the optimization of transition state study, solvent effect is neglected.

==References==
<references>
<ref name="xxx">Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols,http://pubs.acs.org/doi/abs/10.1021/ja00111a016.</ref>

<references>
<ref name="BOND_LENGTH">Bond lengths in organic compounds',Frank H. Allen, Olga Kennard, David G. Watson, Lee Brammer, A. Guy Orpen and Robin Taylor
J. Chem. Soc., Perkin Trans. 2, 1987, S1-S19,http://pubs.rsc.org/en/content/articlepdf/1987/p2/p298700000s1.</ref>

<references/>
<ref name="secondary orbital"/> Ian Fleming, Molecular Orbitals And Organic Chemical Reactions.</ref>

<references/>
<ref name="van_der_waals_radius">van der Waals Volumes and Radii; A. Bondi, J. Phys. Chem., 1964, 68 (3), pp 441–451
DOI: 10.1021/j100785a001.</ref>

<references/>
<ref>Ian Fleming, Molecular Orbitals and Organic Chemical Reactions,ISBN 978-0-470-74658-5</ref>

Rep:Mod-YifanDong

2015-01-30T08:19:34Z

Yd1412: /* Computation of the Transition State Geometry */

=Transition States and Reactivity=
The transition state is a first-order saddle point on the potential energy surface, where it corresponds to a minimum at all points except one. The Hammond–Leffler postulate states that the transition state resembles either reactants or products. Methods used below are based on this postulate.The geometries of the transition states are studied computationally using Gaussview.Different calculation methods are introduced and compared. In this exercise, we cover two types of reactions: the Cope Rearrangement and the Diels-Alder reaction.

==The Cope Rearrangement==
The allowed antiperiplanar and gauche conformations of 1,5-hexadiene are optimized to find the energy-minima and symmetrized to find the point group. The potential energies are calculated using energy optimization to a minimum, the thermochemical data are calculated from a frequency optimization. Usually a combination of OPt+Fre is used.
[[File:Cope REARRANGEMENT.jpg|frame|center|The Cope Rearrangement]]

===Optimization of Reactants and Products===

The conformers of 1,5-hexadiene were optimized to minimum using HF/3-21G method. For each conformer, frequency analysis gives all positive vibrations which indicates it reaches the minimum. The table below shows the conformers and their relative energies. The Gauche 3 conformer is the global minimum because of the attractive interaction between the π orbital and the vinyl protons. However, the energy difference between the gauche3 and anti2 conformations is negligible.<ref name="xxx"/>

{| class="wikitable"
|-
! Conformer !! Structure !! Energy/Hartrees !! Relative Energy/kcal/mol !! Point Group
|-
| Gauche 1|| <jmol><jmolApplet>
<title>Gauche 1</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 1 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68772 || 3.10 || C2
|-
| Gauche 2|| <jmol><jmolApplet>
<title>Gauche 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche2 .mol</uploadedFileContents>
</jmolApplet></jmol> || -231.69167 || 0.62 || C2
|-
| Gauche 3|| <jmol><jmolApplet>
<title>Gauche 3</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 3 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69266 || 0.00 || C1
|-
| Gauche 4||<jmol><jmolApplet>
<title>Gauche 4</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche4 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69153 || 0.71 || C2
|-
| Gauche 5|| <jmol><jmolApplet>
<title>Gauche 5</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE5 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68962 || 1.91 || C1
|-
| Gauche 6|| <jmol><jmolApplet>
<title>Gauche 6</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE6 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68916 || 2.20|| C1
|-
| Anti 2|| <jmol><jmolApplet>
<title>Anti 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 2 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69254 || 0.08 || Ci
|-
| Anti 3|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTI_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68907 || 2.25 || C2h
|-
| Anti 4|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 4 .mol</uploadedFileContents>
</jmolApplet></jmol>||-231.69097 ||1.06 ||C1
|}

The Anti 2 conformer was reoptimized using B3LYP/6-31G(d) method.This method gives a lower energy Anti 2 conformer. 6-31G level gives a higher accuracy than the basis set 3-21G.

HF/3-21G[[File:C6H10_ANTICi_OPT.LOG]]|;
B3LYP/6-31G(d)[[File:ANTI CI 6-31G(d).LOG]]

{| class="wikitable" border="1"
|+ Comparison of different optimisation methods
! Method!! Energy/Hartrees!!Relative energy/kcal/mol !!Convergence !!Jmol
|-
| HF/3-21G|| -231.69254|| 0.08|| Item Value Threshold Converged?
Maximum Force 0.000042 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001524 0.001800 YES
RMS Displacement 0.000749 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| B3LYP/6-31G(d)|| -234.61171 || ||Item Value Threshold Converged?
Maximum Force 0.000049 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001410 0.001800 YES
RMS Displacement 0.000469 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_DFT_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

The table below shows the comparison in bond lengths and bond angles using two methods.

{| class="wikitable" border="1"
|+ Geometry comparison of different optimisation methods
! Parameters!! HF-3-21G!!B3LYP/6-31G!!Literature
|-
| C1=C4|| 1.3162|| 1.3382|| 1.3412
|-
| C4-C6|| 1.5089|| 1.5074|| 1.5077
|-
| C6-C8|| 1.5530|| 1.5549|| 1.5362
|-
| C-H(average)||1.0750||1.0997||1.1077
|-
| C1=C4-C6||124.8||122.0||122.5
|-
| C4-C6-C8||111.4||112.6||111.0
|-
| H-C1=C4||121.9||122.0||120.4
|-
| H-C4-C6||115.5||115.5||118.4
|-
| H-C6-H||107.7||106.7||107.1
|-
| C4-C6-C8-C10(Dihedral angle)||179.99||179.97||179.3
|}
Distances in Å, angles in degrees.

A frequency calculation gives all real and positive vibrational frequencies, which indicates it reaches an energy-minimum. Table below gives the comparison of thermochemistry data using two different optimization methods.

{| class="wikitable" border="1"
|+ Thermochemistry data comparison of different optimisation methods
! Type of energies!! Experimental value(B3LYP/6-31G(d))/Hartrees!!Experimental value(HF/3-21G)/Hartrees
|-
| Sum of electronic and zero-point Energies|| -234.469202||-231.539540
|-
| Sum of electronic and thermal Energies|| -234.461856||-231.532566
|-
| Sum of electronic and thermal Enthalpies||-234.460912||-231.531622
|-
| Sum of electronic and thermal Free Energies|| -234.500773||-231.570913
|-
|}

===Optimisation of the Chair and Boat Transition States===

====Chair Transition State====
In the chair transition structure two optimized C3H5 allyl fragments are positioned approximately 2.2 Å apart, with C2h symmetry. Two different methods are set up to optimize the transition structure: Hartree-Fock(3-21G) where the force constants are computed at the beginning of the calculation and the Redundant Coordinate Editor.

=====Hartree-Fock(3-21G) Method=====

[[File:CHAIR TS GUESS ANIMATION.gif|frame|center|Optimized Chair Transition State Animation using Hartree-Fock Method]]

In this Hartree-Fock method, a Gaussian optimization was set up as Opt+Fre with Optimization to a TS(Berny). The force constant was calculated once. This stops the calculation crashing if more than one imaginary frequency is detected.The calculation gives an imaginary frequency of -818 cm-1 and a energy of -231.61932 Hartrees.

This method is only used when you have a resonable guess for the transition structure. The optimization is repeated using DFT/B3YLP/6-31G method, which gives an electronic energy of -234.55698 Hartrees and an imaginary frequency of -566 cm-1.

This is the one corresponding to the Cope rearrangement.

=====Frozen Coordinate Method=====
This method is used when the guessed structure is far away from the transition structure.

The Redundant Coordinate Editor is used to freeze the coordinates. The distance between two of the terminal carbons of the allyl fragments is fixed to be 2.2 Å. The distance between the opposite two of the terminal carbons of the allyl fragments is also fixed to be 2.2Å.

The optimized chair transition structure is like the sturcture optimized using HF/3-21G method, with an electronic energy of -231.69167 Hartrees and a imaginary frequency of -765 cm-1

The two coordinates we differentiate along the path is shown using numerical normal guess Hessian. The newly formed σ C-C bond distance is 1.5509 Å. Compared with the one optimized above using HF/3-21G, this method gives a longer bond forming length.

[[File:CHAIR FROZEN COORDINATE HESSIAN.gif|frame|center|Optimized Chair Transition State Animation using Frozen Coordinate Method with Modified Heissan.]]

====Boat Transition State====
In the boat transition structure two allyl fragments are positioned 2.2Å apart, with C2v symmetry. The structure is optimized using QST2 method, where the calculation will interpolate between reactants and products and locate a transition state in between them. This method would fail if the reactant and product structures are not close enough to the transition state structure.

[[File:BOAT LABEL ARRANGEMENT.PNG|thumb|center|Reactant and Product with Labelled Atoms |500px]]

The structure above is like a more dissociated version of the chair transition structure. The limitation of QST2 method is that the job is only successful when the input file includes a structure closer to the boat transition structure. In order to optimize the transition state the structure is modified as shown below.

[[File:BOAT ARRANGEMENT2.PNG|thumb|center|Reactant and Product with Labelled Atoms and Modified Arrangements|600px]]

QST3 method is more reliable than QST2 because it allows you to input the guessed transition structure.

=====Intrinsic Reaction Coordinate=====

IRC method calculates the minimum energy path from a transition state structure to the local minimum. Since the reaction is symmetrical, the reaction coordinate is computed only in the forward direction. If given an unsymmetrical reaction then the reaction coordinate will be computed in both directions. As the frequency calculation was previously computed, we calculate the force constants once, rather than at every step along the path. This IRC method gives an electronic energy of -231.69153 Hartrees. The animation is shown below.

[[File:CHAIR IRC VIBRATION.gif|frame|center|Optimized Chair Transition State using IRC method]]

The IRC method didn't give a minimum geometry. It only gives the direction where the slope of the potential energy surface is steepest. The chair transition structure is re-optimized using three approaches listed below.

(1)Minimization for the last point on the IRC
Minimization of the last point on the IRC gives an electronic energy of -231.69167 Hartrees. This approach requires the structure close to the chair transition state, otherwise it will give a wrong minimum.

(2)Change the number of points along the IRC to a higher number until it reaches a minimum.From the table, we can tell that an increase in number of points leads to a alsightly lower energy.

{| class="wikitable" border="1"
|+ Optimization along the IRC
! Number of Points!! Electronic Energy
|-
| 100 || -231.69158 Hartrees
|-
| 150 || -231.69158 Hartrees
|-
| 200 || -231.69158 Hartrees
|}

====Summary of Results====

The table below shows the comparison of activation energies in HF/3-21G and B3LYP/6-31G.
{| class="wikitable" border="1"
|+ Summary of Activation Energies(in kcal/mol)
! Terms!!HF/3-21G!! HF/3-21G!!B3LYP/6-31G!!B3LYP/6-31G!!Experimental Value
|-
! !!at 0K!!at 298.15K!! at 0K!!at 298.15K!!at 0K
|-
| ΔE(chair)|| 45.71||44.70||34.07||33.16||33.5±0.5
|-
| ΔE(boat)|| 55.60||54.76||41.96||41.32||44.7±2.0
|}

The table below shows the thermochemical data for chair and boat transition structures.
{| class="wikitable" border="1"
|+ Summary of Energies(in Hartreers)
! !!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
! !!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies!!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies
|-
! !! !! at 0K!! at 298.15K!!!!at 0K!! at 298.15K
|-
| Chair TS|| -231.61932||-231.46670 || -231.46134||-234.55698 || -234.41491 ||-234.40898
|-
| Boat TS|| -231.60280||-231.45093 ||-231.44530|| -234.54308||-234.40236|| -234.39601
|}

==The Diels Alder Cycloaddition==

Diels-Alder is a [4+2] cycloaddition reaction. Diels-Alder reactions occur between a conjugated s-cis diene and a dienophile. In this exercise, the reaction between maleic anhydride and cyclopentadiene is studied. The less stable product, the endo product is formed in the irreversible Diels-Alder reactions. The kinetic product is formed faster because the bonding interaction in transition state between the electron-deficient C=O groups and back of the diene lowers the energy of its transition state.

The HOMO and LUMO interact when there is a significant overlap between them.They must have the same symmetry for a good overlap. In the cis-butadiene and ethylene cycloaddition, the HOMO of ethylene and the LUMO of cis-butadiene are both symmetric with respect to the plane, the LUMO of ethylene and the HOMO of cis-butadiene are both antisymmetric with respect to the plane.

===Optimization of Cis-Butadiene and Ethylene===
The table below shows the HOMO and LUMO of cis-butadiene and ethylene.The structures are optimized using Hartree-Fock/3-21G method and the MOs are visualized with an isovalue of 0.02.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Transition Structures
!colspan="3"|'''Cis-Butadiene'''
!colspan="3"|'''Ethylene'''
|-
! Term!!Molecular Orbitals!! Symmetry!!Term!!Molecular Orbitals!! Symmetry
|-
| LUMO|| [[File:LUMO of cis-butadiene.PNG]] || Symmetric||LUMO|| [[File:LUMO of ETHYLENE.PNG]]|| Anti Symmetric
|-
| HOMO|| [[File:HOMO of CIS BUTADIENE .PNG]]|| Anti symmetric||HOMO|| [[File:HOMO of ETHYLENE.PNG]] || Symmetric
|}

===Computation of the Transition State Geometry===

[[File:PROTOTYPE TS OPT.gif]]

The Hartree-Fock/3-21G method gives an imaginary frequency of -818 cm-1. AM1 Semi-emipirical method gives an imaginary frequency of -956 cm-1. The presence of an imaginary frequency indicates a negative secondary derivative, which corresponds to the transition state. The animation above corresponds to the reaction path,we can tell from it that the formation of the two bonds is synchronous. From the MOs of the transition states we can tell LUMO of ethylene and HOMO of cis-butadiene interact to give an antisymmetric HOMO.HOMO of ethylene and LUMO of cis-butadiene interact to give a symmetric HOMO.

HOMO and LUMO of the transition states are shown below.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Prototype Reaction Transition Structure
!colspan="4"|'''HF/3-21G'''
!colspan="4"|'''AM1 Semi-Empirical'''
|-
! Term!! Molecular Orbitals!! Symmetry!!Relative Energy!!Term!! Molecular Orbitals!! Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of Prototype REACTION TS.PNG]]||Symmetric||0.14241||LUMO||[[File:LUMO of PROTOTYPE TS OPT AM1.PNG]]||Symmetric||0.02315
|-
| HOMO|| [[File:HOMO of Prototyre Reaction TS.PNG]]|| Symmetric||-0.30087||HOMO||[[File:HOMO of PROTOTYPE TS OPT AM1 .PNG]] ||AntiSymmetric||-0.32394
|}

MOs are visualized with an isovalue of 0.02 in both methods.

A typical sp3 C-C bond distance is 1.54 cm. A typical sp2 C-C bond distance is 1.50 cm.
The van der Waals radius of the C atom vibration is 1.7 Å.
The bond length of the partly formed C-C bond is 2.20 Å which is longer than the normal sp3 C-C bond distance and the van der Waals radius.

===Regioselectivity of the Diels Alder Reaction===
The kinetically controlled reaction between cyclohexa-1,3-diene and maleic anhydride is studied by guessing and optimizing its transition state using the method we introduced above. The corresponding energy for each conformation is calculated using Gaussview. The major product formed is the endo product. The endo rule is explained computationally.

[[File:Diels alder 2.png|thumb|center|Diels Alder Reaction between Cyclohexa-1,3-diene and Maleic Anhydride|500px]]

====HOMO and LUMO of Transition States====

The transition structure is optimized using AM1 Semi-empirical method. The reaction path is shown in the animation below.

{| class="wikitable" border="1"
|+ ENDO and EXO Transition States
! Term!! ENDO!!EXO
|-
| Transition Structure|| [[Image:ENDO TS OPT.gif]] || [[Image:EXO OPT TS.gif]]
|-
| Energy|| -0.051505 Hartrees|| -0.050419 Hartrees
|-
| Imaginary Frequency|| -806 cm-1|| -812 cm-1
|}

The less stable product, the endo product is formed in this irreversible Diels-Alder reactions. Besides the usual primary interaction, there is an additional bonding interaction between the electron-deficient C=O grouphes and back of the diene.This interaction is known as a secondary orbital interaction, which lowers the transition state energy relative to the exo product.<ref name="secondary orbital"/>

[[File:SECONDARY ORBITAL.png|thumb|center|Secondary Orbital Interaction|500px]]

{| class="wikitable" border="1"
|+ HOMO and LUMO of ENDO and EXO Transition States
!colspan="4"|'''ENDO'''
!colspan="4"|'''EXO'''
|-
! Term!! Molecular Orbitals!!Symmetry!!Relative Energy!!Term!! Molecular Orbitals!!Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of ENDO.PNG ]]||Antisymmetric || -0.03570 || LUMO || [[File:EXO LUMO .PNG ]] || Antisymmetric || 0.00601
|-
| HOMO|| [[File:HOMO of ENDO.PNG ]]|| Antisymmetric || -0.34505 || HOMO || [[File:EXO HOMO .PNG ]] ||Symmetric || -0.38787
|}

====Intrinsic Reaction Coordinate Calculation====

{| class="wikitable" border="1"
|+ IRC and RMS Gradient of ENDO and EXO Conformations
! Term!! ENDO!!EXO
|-
| IRC path|| [[File:ENDO IRC .PNG]] || [[File:EXO IRC .PNG]]
|-
| RMS Gradient||[[File:ENDO RMS.PNG]] || [[File:EXO RMS.PNG]]
|-
| Relative Energy of Transition State/Hartrees|| -0.051505 ||-0.050420
|-
|Relative Energy of Product at IRC=-5.9/Hartrees||-0.15985||-0.15991
|}

AM1 method gives an IRC path which resembles the path going from product(on the left) to reactants(on the right). The reactants tend to be far away from each other therefore the calculation does not converge.A weird point on the IRC curve is observed after the transition state, where it suddenly drops to the product. However, this point does not show up in the corresponding RMS gradient curve.
IRC curve proves the optimized transition structures for endo and exo products are correct. A correct optimized transition structure has a gradient of 0.

===Further Discussion===
In the optimization of transition state study, solvent effect is neglected.

==References==
<references>
<ref name==="xxx">Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols,http://pubs.acs.org/doi/abs/10.1021/ja00111a016.</ref>

<references>
<ref name="BOND_LENGTH">'Bond lengths in organic compounds',Frank H. Allen, Olga Kennard, David G. Watson, Lee Brammer, A. Guy Orpen and Robin Taylor
J. Chem. Soc., Perkin Trans. 2, 1987, S1-S19,http://pubs.rsc.org/en/content/articlepdf/1987/p2/p298700000s1.</ref>

<references/>
<ref name="secondary orbital"/> Ian Fleming, Molecular Orbitals And Organic Chemical Reactions.</ref>

<references/>
<ref name="van_der_waals_radius">van der Waals Volumes and Radii; A. Bondi, J. Phys. Chem., 1964, 68 (3), pp 441–451
DOI: 10.1021/j100785a001.</ref>

<ref>ISBN 978-0-470-74658-5</ref><ref></ref>

Rep:Mod-YifanDong

2015-01-30T08:14:46Z

Yd1412: /* Transition States and Reactivity */

=Transition States and Reactivity=
The transition state is a first-order saddle point on the potential energy surface, where it corresponds to a minimum at all points except one. The Hammond–Leffler postulate states that the transition state resembles either reactants or products. Methods used below are based on this postulate.The geometries of the transition states are studied computationally using Gaussview.Different calculation methods are introduced and compared. In this exercise, we cover two types of reactions: the Cope Rearrangement and the Diels-Alder reaction.

==The Cope Rearrangement==
The allowed antiperiplanar and gauche conformations of 1,5-hexadiene are optimized to find the energy-minima and symmetrized to find the point group. The potential energies are calculated using energy optimization to a minimum, the thermochemical data are calculated from a frequency optimization. Usually a combination of OPt+Fre is used.
[[File:Cope REARRANGEMENT.jpg|frame|center|The Cope Rearrangement]]

===Optimization of Reactants and Products===

The conformers of 1,5-hexadiene were optimized to minimum using HF/3-21G method. For each conformer, frequency analysis gives all positive vibrations which indicates it reaches the minimum. The table below shows the conformers and their relative energies. The Gauche 3 conformer is the global minimum because of the attractive interaction between the π orbital and the vinyl protons. However, the energy difference between the gauche3 and anti2 conformations is negligible.<ref name="xxx"/>

{| class="wikitable"
|-
! Conformer !! Structure !! Energy/Hartrees !! Relative Energy/kcal/mol !! Point Group
|-
| Gauche 1|| <jmol><jmolApplet>
<title>Gauche 1</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 1 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68772 || 3.10 || C2
|-
| Gauche 2|| <jmol><jmolApplet>
<title>Gauche 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche2 .mol</uploadedFileContents>
</jmolApplet></jmol> || -231.69167 || 0.62 || C2
|-
| Gauche 3|| <jmol><jmolApplet>
<title>Gauche 3</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche 3 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69266 || 0.00 || C1
|-
| Gauche 4||<jmol><jmolApplet>
<title>Gauche 4</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Gauche4 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69153 || 0.71 || C2
|-
| Gauche 5|| <jmol><jmolApplet>
<title>Gauche 5</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE5 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68962 || 1.91 || C1
|-
| Gauche 6|| <jmol><jmolApplet>
<title>Gauche 6</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>GAUCHE6 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68916 || 2.20|| C1
|-
| Anti 2|| <jmol><jmolApplet>
<title>Anti 2</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 2 .mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.69254 || 0.08 || Ci
|-
| Anti 3|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTI_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>|| -231.68907 || 2.25 || C2h
|-
| Anti 4|| <jmol><jmolApplet>
<title>test molecule</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>Anti 4 .mol</uploadedFileContents>
</jmolApplet></jmol>||-231.69097 ||1.06 ||C1
|}

The Anti 2 conformer was reoptimized using B3LYP/6-31G(d) method.This method gives a lower energy Anti 2 conformer. 6-31G level gives a higher accuracy than the basis set 3-21G.

HF/3-21G[[File:C6H10_ANTICi_OPT.LOG]]|;
B3LYP/6-31G(d)[[File:ANTI CI 6-31G(d).LOG]]

{| class="wikitable" border="1"
|+ Comparison of different optimisation methods
! Method!! Energy/Hartrees!!Relative energy/kcal/mol !!Convergence !!Jmol
|-
| HF/3-21G|| -231.69254|| 0.08|| Item Value Threshold Converged?
Maximum Force 0.000042 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001524 0.001800 YES
RMS Displacement 0.000749 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| B3LYP/6-31G(d)|| -234.61171 || ||Item Value Threshold Converged?
Maximum Force 0.000049 0.000450 YES
RMS Force 0.000008 0.000300 YES
Maximum Displacement 0.001410 0.001800 YES
RMS Displacement 0.000469 0.001200 YES
||<jmol><jmolApplet>
<title>Anti_Ci</title>
<color>lightblue</color>
<size>200</size>
<uploadedFileContents>C6H10_ANTICi_DFT_OPT.mol</uploadedFileContents>
</jmolApplet></jmol>
|}

The table below shows the comparison in bond lengths and bond angles using two methods.

{| class="wikitable" border="1"
|+ Geometry comparison of different optimisation methods
! Parameters!! HF-3-21G!!B3LYP/6-31G!!Literature
|-
| C1=C4|| 1.3162|| 1.3382|| 1.3412
|-
| C4-C6|| 1.5089|| 1.5074|| 1.5077
|-
| C6-C8|| 1.5530|| 1.5549|| 1.5362
|-
| C-H(average)||1.0750||1.0997||1.1077
|-
| C1=C4-C6||124.8||122.0||122.5
|-
| C4-C6-C8||111.4||112.6||111.0
|-
| H-C1=C4||121.9||122.0||120.4
|-
| H-C4-C6||115.5||115.5||118.4
|-
| H-C6-H||107.7||106.7||107.1
|-
| C4-C6-C8-C10(Dihedral angle)||179.99||179.97||179.3
|}
Distances in Å, angles in degrees.

A frequency calculation gives all real and positive vibrational frequencies, which indicates it reaches an energy-minimum. Table below gives the comparison of thermochemistry data using two different optimization methods.

{| class="wikitable" border="1"
|+ Thermochemistry data comparison of different optimisation methods
! Type of energies!! Experimental value(B3LYP/6-31G(d))/Hartrees!!Experimental value(HF/3-21G)/Hartrees
|-
| Sum of electronic and zero-point Energies|| -234.469202||-231.539540
|-
| Sum of electronic and thermal Energies|| -234.461856||-231.532566
|-
| Sum of electronic and thermal Enthalpies||-234.460912||-231.531622
|-
| Sum of electronic and thermal Free Energies|| -234.500773||-231.570913
|-
|}

===Optimisation of the Chair and Boat Transition States===

====Chair Transition State====
In the chair transition structure two optimized C3H5 allyl fragments are positioned approximately 2.2 Å apart, with C2h symmetry. Two different methods are set up to optimize the transition structure: Hartree-Fock(3-21G) where the force constants are computed at the beginning of the calculation and the Redundant Coordinate Editor.

=====Hartree-Fock(3-21G) Method=====

[[File:CHAIR TS GUESS ANIMATION.gif|frame|center|Optimized Chair Transition State Animation using Hartree-Fock Method]]

In this Hartree-Fock method, a Gaussian optimization was set up as Opt+Fre with Optimization to a TS(Berny). The force constant was calculated once. This stops the calculation crashing if more than one imaginary frequency is detected.The calculation gives an imaginary frequency of -818 cm-1 and a energy of -231.61932 Hartrees.

This method is only used when you have a resonable guess for the transition structure. The optimization is repeated using DFT/B3YLP/6-31G method, which gives an electronic energy of -234.55698 Hartrees and an imaginary frequency of -566 cm-1.

This is the one corresponding to the Cope rearrangement.

=====Frozen Coordinate Method=====
This method is used when the guessed structure is far away from the transition structure.

The Redundant Coordinate Editor is used to freeze the coordinates. The distance between two of the terminal carbons of the allyl fragments is fixed to be 2.2 Å. The distance between the opposite two of the terminal carbons of the allyl fragments is also fixed to be 2.2Å.

The optimized chair transition structure is like the sturcture optimized using HF/3-21G method, with an electronic energy of -231.69167 Hartrees and a imaginary frequency of -765 cm-1

The two coordinates we differentiate along the path is shown using numerical normal guess Hessian. The newly formed σ C-C bond distance is 1.5509 Å. Compared with the one optimized above using HF/3-21G, this method gives a longer bond forming length.

[[File:CHAIR FROZEN COORDINATE HESSIAN.gif|frame|center|Optimized Chair Transition State Animation using Frozen Coordinate Method with Modified Heissan.]]

====Boat Transition State====
In the boat transition structure two allyl fragments are positioned 2.2Å apart, with C2v symmetry. The structure is optimized using QST2 method, where the calculation will interpolate between reactants and products and locate a transition state in between them. This method would fail if the reactant and product structures are not close enough to the transition state structure.

[[File:BOAT LABEL ARRANGEMENT.PNG|thumb|center|Reactant and Product with Labelled Atoms |500px]]

The structure above is like a more dissociated version of the chair transition structure. The limitation of QST2 method is that the job is only successful when the input file includes a structure closer to the boat transition structure. In order to optimize the transition state the structure is modified as shown below.

[[File:BOAT ARRANGEMENT2.PNG|thumb|center|Reactant and Product with Labelled Atoms and Modified Arrangements|600px]]

QST3 method is more reliable than QST2 because it allows you to input the guessed transition structure.

=====Intrinsic Reaction Coordinate=====

IRC method calculates the minimum energy path from a transition state structure to the local minimum. Since the reaction is symmetrical, the reaction coordinate is computed only in the forward direction. If given an unsymmetrical reaction then the reaction coordinate will be computed in both directions. As the frequency calculation was previously computed, we calculate the force constants once, rather than at every step along the path. This IRC method gives an electronic energy of -231.69153 Hartrees. The animation is shown below.

[[File:CHAIR IRC VIBRATION.gif|frame|center|Optimized Chair Transition State using IRC method]]

The IRC method didn't give a minimum geometry. It only gives the direction where the slope of the potential energy surface is steepest. The chair transition structure is re-optimized using three approaches listed below.

(1)Minimization for the last point on the IRC
Minimization of the last point on the IRC gives an electronic energy of -231.69167 Hartrees. This approach requires the structure close to the chair transition state, otherwise it will give a wrong minimum.

(2)Change the number of points along the IRC to a higher number until it reaches a minimum.From the table, we can tell that an increase in number of points leads to a alsightly lower energy.

{| class="wikitable" border="1"
|+ Optimization along the IRC
! Number of Points!! Electronic Energy
|-
| 100 || -231.69158 Hartrees
|-
| 150 || -231.69158 Hartrees
|-
| 200 || -231.69158 Hartrees
|}

====Summary of Results====

The table below shows the comparison of activation energies in HF/3-21G and B3LYP/6-31G.
{| class="wikitable" border="1"
|+ Summary of Activation Energies(in kcal/mol)
! Terms!!HF/3-21G!! HF/3-21G!!B3LYP/6-31G!!B3LYP/6-31G!!Experimental Value
|-
! !!at 0K!!at 298.15K!! at 0K!!at 298.15K!!at 0K
|-
| ΔE(chair)|| 45.71||44.70||34.07||33.16||33.5±0.5
|-
| ΔE(boat)|| 55.60||54.76||41.96||41.32||44.7±2.0
|}

The table below shows the thermochemical data for chair and boat transition structures.
{| class="wikitable" border="1"
|+ Summary of Energies(in Hartreers)
! !!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
! !!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies!!Electronic energy!!Sum of electronic and zero-point energies!!Sum of electronic and thermal energies
|-
! !! !! at 0K!! at 298.15K!!!!at 0K!! at 298.15K
|-
| Chair TS|| -231.61932||-231.46670 || -231.46134||-234.55698 || -234.41491 ||-234.40898
|-
| Boat TS|| -231.60280||-231.45093 ||-231.44530|| -234.54308||-234.40236|| -234.39601
|}

==The Diels Alder Cycloaddition==

Diels-Alder is a [4+2] cycloaddition reaction. Diels-Alder reactions occur between a conjugated s-cis diene and a dienophile. In this exercise, the reaction between maleic anhydride and cyclopentadiene is studied. The less stable product, the endo product is formed in the irreversible Diels-Alder reactions. The kinetic product is formed faster because the bonding interaction in transition state between the electron-deficient C=O groups and back of the diene lowers the energy of its transition state.

The HOMO and LUMO interact when there is a significant overlap between them.They must have the same symmetry for a good overlap. In the cis-butadiene and ethylene cycloaddition, the HOMO of ethylene and the LUMO of cis-butadiene are both symmetric with respect to the plane, the LUMO of ethylene and the HOMO of cis-butadiene are both antisymmetric with respect to the plane.

===Optimization of Cis-Butadiene and Ethylene===
The table below shows the HOMO and LUMO of cis-butadiene and ethylene.The structures are optimized using Hartree-Fock/3-21G method and the MOs are visualized with an isovalue of 0.02.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Transition Structures
!colspan="3"|'''Cis-Butadiene'''
!colspan="3"|'''Ethylene'''
|-
! Term!!Molecular Orbitals!! Symmetry!!Term!!Molecular Orbitals!! Symmetry
|-
| LUMO|| [[File:LUMO of cis-butadiene.PNG]] || Symmetric||LUMO|| [[File:LUMO of ETHYLENE.PNG]]|| Anti Symmetric
|-
| HOMO|| [[File:HOMO of CIS BUTADIENE .PNG]]|| Anti symmetric||HOMO|| [[File:HOMO of ETHYLENE.PNG]] || Symmetric
|}

===Computation of the Transition State Geometry===

[[File:PROTOTYPE TS OPT.gif]]

The Hartree-Fock/3-21G method gives an imaginary frequency of -818 cm-1. AM1 Semi-emipirical method gives an imaginary frequency of -956 cm-1. The presence of an imaginary frequency indicates a negative secondary derivative, which corresponds to the transition state. The animation above corresponds to the reaction path,we can tell from it that the formation of the two bonds is synchronous. From the MOs of the transition states we can tell LUMO of ethylene and HOMO of cis-butadiene interact to give an antisymmetric HOMO.HOMO of ethylene and LUMO of cis-butadiene interact to give a symmetric HOMO.

HOMO and LUMO of the transition states are shown below.

{| class="wikitable" border="1"
|+ HOMO and LUMO of Prototype Reaction Transition Structure
!colspan="4"|'''HF/3-21G'''
!colspan="4"|'''AM1 Semi-Empirical'''
|-
! Term!! Molecular Orbitals!! Symmetry!!Relative Energy!!Term!! Molecular Orbitals!! Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of Prototype REACTION TS.PNG]]||Symmetric||0.14241||LUMO||[[File:LUMO of PROTOTYPE TS OPT AM1.PNG]]||Symmetric||0.02315
|-
| HOMO|| [[File:HOMO of Prototyre Reaction TS.PNG]]|| Symmetric||-0.30087||HOMO||[[File:HOMO of PROTOTYPE TS OPT AM1 .PNG]] ||AntiSymmetric||-0.32394
|}

MOs are visualized with an isovalue of 0.02 in both methods.

A typical sp3 C-C bond distance is 1.54 cm. A typical sp2 C-C bond distance is 1.50 cm.<ref name="BOND_LENGTH"/> The van der Waals radius of the C atom vibration is 1.7 Å.<ref name="van_der_waals_radius">The bond length of the partly formed C-C bond is 2.20 Å which is longer than the normal sp3 C-C bond distance and the van der Waals radius.

===Regioselectivity of the Diels Alder Reaction===
The kinetically controlled reaction between cyclohexa-1,3-diene and maleic anhydride is studied by guessing and optimizing its transition state using the method we introduced above. The corresponding energy for each conformation is calculated using Gaussview. The major product formed is the endo product. The endo rule is explained computationally.

[[File:Diels alder 2.png|thumb|center|Diels Alder Reaction between Cyclohexa-1,3-diene and Maleic Anhydride|500px]]

====HOMO and LUMO of Transition States====

The transition structure is optimized using AM1 Semi-empirical method. The reaction path is shown in the animation below.

{| class="wikitable" border="1"
|+ ENDO and EXO Transition States
! Term!! ENDO!!EXO
|-
| Transition Structure|| [[Image:ENDO TS OPT.gif]] || [[Image:EXO OPT TS.gif]]
|-
| Energy|| -0.051505 Hartrees|| -0.050419 Hartrees
|-
| Imaginary Frequency|| -806 cm-1|| -812 cm-1
|}

The less stable product, the endo product is formed in this irreversible Diels-Alder reactions. Besides the usual primary interaction, there is an additional bonding interaction between the electron-deficient C=O grouphes and back of the diene.This interaction is known as a secondary orbital interaction, which lowers the transition state energy relative to the exo product.<ref name="secondary orbital"/>

[[File:SECONDARY ORBITAL.png|thumb|center|Secondary Orbital Interaction|500px]]

{| class="wikitable" border="1"
|+ HOMO and LUMO of ENDO and EXO Transition States
!colspan="4"|'''ENDO'''
!colspan="4"|'''EXO'''
|-
! Term!! Molecular Orbitals!!Symmetry!!Relative Energy!!Term!! Molecular Orbitals!!Symmetry!!Relative Energy
|-
| LUMO|| [[File:LUMO of ENDO.PNG ]]||Antisymmetric || -0.03570 || LUMO || [[File:EXO LUMO .PNG ]] || Antisymmetric || 0.00601
|-
| HOMO|| [[File:HOMO of ENDO.PNG ]]|| Antisymmetric || -0.34505 || HOMO || [[File:EXO HOMO .PNG ]] ||Symmetric || -0.38787
|}

====Intrinsic Reaction Coordinate Calculation====

{| class="wikitable" border="1"
|+ IRC and RMS Gradient of ENDO and EXO Conformations
! Term!! ENDO!!EXO
|-
| IRC path|| [[File:ENDO IRC .PNG]] || [[File:EXO IRC .PNG]]
|-
| RMS Gradient||[[File:ENDO RMS.PNG]] || [[File:EXO RMS.PNG]]
|-
| Relative Energy of Transition State/Hartrees|| -0.051505 ||-0.050420
|-
|Relative Energy of Product at IRC=-5.9/Hartrees||-0.15985||-0.15991
|}

AM1 method gives an IRC path which resembles the path going from product(on the left) to reactants(on the right). The reactants tend to be far away from each other therefore the calculation does not converge.A weird point on the IRC curve is observed after the transition state, where it suddenly drops to the product. However, this point does not show up in the corresponding RMS gradient curve.
IRC curve proves the optimized transition structures for endo and exo products are correct. A correct optimized transition structure has a gradient of 0.

===Further Discussion===
In the optimization of transition state study, solvent effect is neglected.

==References==
<references>
<ref name==="xxx">Conformational Study of 1,5-Hexadiene and 1,5-Diene-3,4-diols,http://pubs.acs.org/doi/abs/10.1021/ja00111a016.</ref>

<references>
<ref name="BOND_LENGTH">'Bond lengths in organic compounds',Frank H. Allen, Olga Kennard, David G. Watson, Lee Brammer, A. Guy Orpen and Robin Taylor
J. Chem. Soc., Perkin Trans. 2, 1987, S1-S19,http://pubs.rsc.org/en/content/articlepdf/1987/p2/p298700000s1.</ref>

<references/>
<ref name="secondary orbital"/> Ian Fleming, Molecular Orbitals And Organic Chemical Reactions.</ref>

<references/>
<ref name="van_der_waals_radius">van der Waals Volumes and Radii; A. Bondi, J. Phys. Chem., 1964, 68 (3), pp 441–451
DOI: 10.1021/j100785a001.</ref>

<ref>ISBN 978-0-470-74658-5</ref><ref></ref>