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<h1 align="center">Module 3: Charlene Lawton</h1>
<h2>The Cope Rearrangement of 1,5-hexadiene</h2>

The low-energy minima and transition states on the potential energy surface (PES) of 1,5-hexadiene were investigated to determine the preferred reaction mechanism. The [3,3]-sigmatropic shift rearrangement (shown below) occurs in a concerted fashion passing through a chair or boat transition state (TS).

[[Image:Cope_rearrangement_reactncfl08.jpg|centre|The Cope Rearrangement and TS of 1,5-hexadiene|600x203px]]

Optimizing the Reactants and Products

The geometry of 1,5-hexadiene in an anti-periplanar (APP) arrangement and in a gauche arrangement were optimised at the HF/3-21G level of theory. This gave the results tabulated below.

{| class="wikitable" border="1" align="left"
|+ ''Optimised Gauche and APP 1,5-hexadiene conformers''
!!!<jmol><jmolAppletButton><uploadedFileContents>React_anticfl08.mol</uploadedFileContents><text>APP1</text></jmolAppletButton> </jmol>!!<jmol><jmolAppletButton><uploadedFileContents>Gauche hexadienecfl08.mol</uploadedFileContents><text>Gauche</text></jmolAppletButton></jmol>
|-
| Point Group || C2 || C1
|-
| Energy/ Hartree|| -231.69260 || -231.69266
|}
 
It was expected that the gauche linkage would be less stable than the anti linkage; however the computed energies above show the gauche model to be more stable. This is because the additional H-H contacts lead to the gecko effect where the Van der Waals attractive interaction makes the gauche conformation more favourable (ref rzepa). Also there are six possible gauche conformers and only four possible APP conformers meaning the gauche arrangement is entropically favoured. The gauche structure was compared to the reference table and it was confirmed that this conformer is the lowest energy minima. The energy of the APP structure also agrees with the reference energy given in the table. link ref table

Another APP conformer (APP2) of 1,5-hexadiene was optimised at the HF/3-21G level of theory; this gave a structure with Ci symmetry and an energy of -231.69254 Hartree. This structure was then re-optimised to B3LYP/6-31G* level to give a structure with the same point group and an energy of -234.62318 Hartree. The structures look similar - both have a dihedral angle of 180 degrees. The carbon-carbon single bond lengths are slightly shorter whilst the carbon-carbon double bond is slightly longer in the structure optimised to a higher level c.f. the table below.

{| class="wikitable" border="1" align="left"
|+ ''Comparing the optimised APP2 models''
!Bond!!APP2 - HF/3-21G!!APP2 - B3LYP/6-31G*
|-
|C=C||1.32||1.33
|-
|Central C-C || 1.55 || 1.53
|-
|C-C || 1.51 || 1.48
|}
 

<jmol><jmolAppletButton><uploadedFileContents>APP 2cfl08.mol</uploadedFileContents><text>APP2 - HF/3-21G</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>APP 2 opt 2cfl08.mol</uploadedFileContents><text>APP2 - B3LYP/6-31G*</text></jmolAppletButton></jmol>

Vibrational frequencies are the second derivative of the potential energy and therefore minimal energy structures can be confirmed by vibrational analysis; if vibrational frequencies are real and positive this corresponds to a minimum point.

For the higher level calculation: 
Low frequencies: - 38.2274, - 14.7195, 0.0006, 0.0009, 0.0010, 9.6799 
Low frequencies: 73.4693, 74.1196, 119.5657

IR spectrum of the higher level calculation 
[[Image:APP2_IR_speccfl08.jpeg]]

The vibrational analysis computes real properties:
Sum of electronic and zero-point Energies= -233.125690
Sum of electronic and thermal Energies= -233.118189
Sum of electronic and thermal Enthalpies= -233.117245
Sum of electronic and thermal Free Energies= -233.157407

The sum of electronic and zero-point energies is the potential energy at 0 K including the zero-point vibrational energy (E = Eelec + ZPE). The sum of electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans. The sum of electronic and thermal enthalpies contains an additional correction for RT (H = E + RT) which is important when looking at dissociation reactions, and the last sum listed above includes the entropic contribution to the free energy (G = H - TS).

<h2>Optimising the Boat and Chair Transition Structures</h2>

The two possible transition structures (boat or chair) were optimised by 
(1) by computing the force constants at the beginning of the calculation 
(2) using the redundant coordinate editor, and 
(3) using QST2.
The reaction coordinate was visualised, the IRC was run and the activation energies for the Cope rearrangement.

<h3>Chair TS optimisation</h3>

<h4>Method: Computing the force constant matrix</h4>

Initially the delocalised allyl fragment (CH2CHCH2) was optimised (HF/3-21G) and two of these fragments were positioned in the assumed chair transition state with the two terminal carbons approximately 2.2 angstroms apart. The TS was optimised to a TS Berny by computing the force constant matrix (also known as the Hessian) in the first step of the optimization which was then be updated as the optimization proceeded. The opt+freq job type, Hartree Fock method, and the default basis set 3-21G were selected; the force constants were chosen to be calculated once with the additional keywords "Opt=NoEigen" to stop the calculation crashing if more than one imaginary frequency had been discovered. The optimised TS has a distance of 2.02 angstrom between the terminal carbons of each fragment. The frequency analysis showed one imaginary frequency at which corresponds to the Cope rearrangement at -818 cm-1 (0dp).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents>Chair ts guess opt and freqcfl08.mol</uploadedFileContents><text>Chair TS</text></jmolAppletButton></jmol>
 Chair TS Vibration corresponding to the Cope Rearrangement 
[[Image:Chair_TS_cope_vibrationcfl08.gif]]

<h4>Method: Using the redundant coordinate editor </h4>

The TS was then optimised by freezing the reaction coordinate, using Opt=ModRedundant, and minimizing the rest of the molecule. Once the molecule is fully relaxed, the reaction coordinate is unfrozen and the TS optimization is started again. This method means that it may not be necessary to compute the whole Hessian once this has been done, and just differentiating along the reaction coordinate which can save time. Note again the Hartree Fock method and 3-21G basis set were used.
The transition state optimisation was run with the frozen coordinates fixed (distance between the terminal carbons where a bond forms/breaks a bond during the rearrangement fixed to 2.2 angstroms), then repeated with the frozen coordinate bond as a derivative (no force constants calculated). This gave optimised distance between the two fragments as 2.02 angstroms which is in agreement with the TS (Berny) method, shown right; the two methods gave the same result.

<jmol><jmolAppletButton><uploadedFileContents>TS chair freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - fixed bond length</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>TS chair bond freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - optimising bond length</text></jmolAppletButton></jmol>

<h3>Boat TS optimisation</h3>

<h4>Method: The QST2</h4>

The boat TS was optimised using the QST2 method where the reactants and products are specified and the calculation interpolates between the two structures to find the TS between them. The numbering of the carbon and hydrogen atoms were manually changed so that they were numbered in the same way.

Labelled reactant (left) and product (right).

[[Image:Labelled_prod_and_reactantcfl.jpeg]]

The fist QTS2 structure did not give the correct structure - this is because the calculation interpolated linearly between the two structures and translated the top allyl fragment to give a dissociated Chair TS. It was also necessary to manually change the orientation of the reactants and products so that the possibility a rotation around the central bonds was considered. For both the product and reactant, the central C-C-C-C dihedral angle was modified to 0 degrees, then the inside C-C-C bonds were reduced to 100 degrees.

Product and reactants with modified angles.

[[Image:Qst2_modified_angles_inputcfl.jpeg]]

The above input was then optimised to a TS (QST2) using the Hartree Fock method and 3-21G basis set, this gave the TS with C-C distance of 2.14, 2.78 and 2.14 angstrom between the two fragments so that the middle carbons on each fragment are further apart. The optimised TS shows one imaginary frequency at -840 cm-1(0 d.p.).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents> Boat qst2 optcfl08.mol</uploadedFileContents><text>Boat TS</text></jmolAppletButton></jmol>
 Boat TS Vibration corresponding to the Cope Rearrangement 
[[Image:Boat_qst2_opt_cope_vibrationcfl.gif]]

<h4>Method: Intrinsic Reaction Coordinate (IRC)</h4>

The IRC method can be used to predict what conformers of 1,5-hexadiene will develop from the optimised chair and boat transition structures. This method allows the minimum energy path,from a transition structure down to its local minimum on a potential energy surface, to be followed. This creates a series of points by taking small geometry steps in the direction where the gradient or slope of the energy surface is steepest.

Force constants were determined once at the start of the calculation, and the reaction coordinate was ran in just the forward direction (since this is symmetrical), with 50 points. Initially a minimum is not reached; the gradient does not fall to zero. There are three ways a minimum can be reach: 
(1) a normal minimisation can be run on the last point on the IRC, 
(2) the IRC can be restarted with a large number of specified points until a minimum is reached 
(3) the IRC can be re-done with force constants computed at every step.
 Although option three is not always feasible for large systems, it is known to be the most reliable therefore this option was chosen. IRC showed the chair TS to result in the gauche 2 structure (see Image below) with an energy of - 231.69166 au, the paths are illustrated graphically below.
 
Chair TS IRC leads to the Gauche 2 conformer 
[[Image:Irc_gauche_2cfl.jpg]] 
Graph 1: IRC method (force constants computed once) - minimisation not yet reached. 
[[Image:Irc_min_not_reachedcfl.jpeg|centre|IRC - min not reached|542x416px]]
 
Graph 2: IRC method (force constants always computed) - minimisation reached. 
[[Image:Irc_min_reachedcfl.jpg|centre|IRC - min reached|542x416px]] 

<h4>Calculating Activation Energies: re-optimising Boat and Chair TS to higher level of theory</h4>

The activation energies for the reaction via both transition states were calculated by re-optimising the chair and boat transition structures using B3LYP/6-31G* level of theory and carrying out a frequency analysis.
Again one imaginary frequency which connects products to reactants was detected for each structure, showing that the structure is a TS.

Chair TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Chair_TS_reoptcfl08.gif]] 
Boat TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Boat_reopt_507cfl.gif]]

{| class="wikitable" border="1" align="left"
|+ ''Comparing HF/3-21G and B3LYP/6-31G* optimised Boat and Chair TS''
!!!Chair HF/3-21G !!Chair B3LYP/6-31G*!!Boat HF/3-21G !! Boat B3LYP/6-31G*
|-
| Energy/ au|| -231.61932 ||-234.56752||-231.60280 || -234.55445
|-
| C-C distance through space between fragments/angstrom|| 2.02||1.98 ||2.14 ||2.24
|-
| Imaginary Frequency/cm-1 || -818 || -563 ||-840 || -507
|}
 

The geometries are similar but the there is a large energy difference between the two levels of theory, approximately 3 Hartree (~ Kcalmol-1. As a result, usually the potential energy surface is mapped at a low theory level and then re-optimised at a higher level to provide more efficient calculations.

The activation energies at 0 Kelvin for the cope rearrangement were calculated by taking the difference between the energy of the reactant and the energy of the transition state.

{| class="wikitable" border="1" align="centre"
|+ ''Activation Energies at 0K (kcal/mol)''
! !! HF/3-21G !! B3LYP/6-31G* !! Experiment
|-
| Chair||45.4 || 35.0 || 33±0.5
|-
|Boat||58.1 || 43.1 || 44.7±2.0
|}

As the calculation is down to a higher level of theory, the activation energy values approach the experimental values. In order to completely match the experimental data an even higher level of theory would have to be used- the observed difference is due to the approximations made in the calculation.

<h1>The Diels Alder Cycloaddition</h1>

<h2>Exercise 1: MOs of Reactants</h2>
Cis-butadiene and ethylene were optimised using the semi-empirical method (AM1).
<jmol><jmolAppletButton><uploadedFileContents>BUTADIENE_OPTcfl08.mol</uploadedFileContents><text>Cis-butadiene</text></jmolAppletButton></jmol>
<jmol><jmolAppletButton><uploadedFileContents>ETHYLENE_OPTcfl08.mol</uploadedFileContents><text>Ethylene</text></jmolAppletButton> </jmol>
 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Reactants''
! Cis-Butadiene LUMO !! Cis-butadiene HOMO !! Ethylene LUMO !! Ethylene HOMO
|-
| [[Image:Cis_butadiene_LUMO.cfljpg]] || [[Image:Cis_butadiene_HOMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_LUMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_HOMOcfl.jpg|centre|HOMO B|244x190px]]
|-
| symmetric || antisymmetric || antisymmetric|| symmetric
|}
 

'''Discussion of MOs'''

The [4s + 2s] pericyclic reaction between ethylene and cis-butadiene depends on the HOMO/LUMO interactions of the molecules. The concerted reaction is allowed if the HOMO of one reactant can interact with the LUMO of the other reactant and when the HOMO-LUMO interaction has significant overlap density. There is no overlap density if the orbitals have different symmetry properties.
The HOMO of ethylene and the LUMO of butadiene are both symmetric with respect to the reflection plane and the LUMO of ethylene and the HOMO of butadiene are both antisymmetric. Therefore, the HOMO-LUMO pairs can interact and the principal orbital interactions involve the π/ π* orbitals of ethylene and the HOMO/LUMO of butadiene; π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene.

<h2>Exercise 1: Computation of the Transition State geometry</h2>

The transition state was optimised using the redundant coordinate editor (Opt=ModRedundant) - the reaction coordinate was frozen and the rest of the molecule minimised (distance between ethylene and butadiene set to 2.2 angstrom) then reaction coordinate was unfrozen and the structure minimised to a TS(Berny). The semi-empirical AM1 method was used.
<jmol><jmolAppletButton><uploadedFileContents>But_eth_TScfl08.mol</uploadedFileContents><text>Optimised Transition State</text></jmolAppletButton></jmol>

 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Optimised Transition State''
!HOMO-1!!HOMO !!LUMO !! LUMO-1
|-
| [[Image:But_eth_TS_HOMO-1cfl.jpeg]] || [[Image:But_eth_TS_HOMO_scfl.jpeg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO_scfl.jpg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO+1cfl.jpeg|centre|HOMO B|224x263px]]
|-
| symmetric || antisymmetric || symmetric|| antisymmetric
|}
 

'''Discussion of TS Geometry and MOs'''

The antisymmetric HOMO of the transition structure, shown above, is formed from the HOMO of cis-butadiene interacting with the LUMO of ethylene. This reaction is allowed beaucse the two interacting orbitals have the same symmetry and significant overlap density.

Below is the optimsied TS with the bond lengths labelled 
[[Image:Diels_Alder_TS_bondlengthscfl08.jpg]]

Typically,

'''Disucssion Vibrational Analysis of TS'''

The structure was confirmed to be a TS by frequency analysis; the frequency analysis showed one imaginary frequency at -956cm-1 which shows the synchronous formation of the two bonds(the reaction is concerted).

 Transition State Imaginary Frequency [[Image:Diels_alder_TS_b2_freqcfl.gif‎‎]]
 Transition State Lowest positive frequency [[Image:]]

Notably the LUMO of the transition state is formed by the combination of the ethylene HOMO and cis-butadiene LUMO (both symmetric, s); the ethylene HOMO and LUMO are of course the pi and pi* bonding and anti-bonding MOs.
Which MOs of butadiene and ethylene have been used to form this MO? Explain why the reaction is allowed.
The π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene and the resulting HOMO is asymmetric.

What are typical sp3 and sp2 C-C bondlengths? What is the van der Waals radius of the C atom? What can you conclude about the C-C bondlength of the partly formed σ C-C bonds in the TS.

ypically sp3 bonded carbon has a C-C bondlength of 1.54Å, and sp2 carbon a C=C bondlength of 1.34Å.[1] The observed bond lengths are intermittant between the two, indicative of a transition state with delocalised bonding. In addition, carbon has an atomic van der Waals radius of 1.70Å.[2] thus non-bonded carbons must be 3.4Å apart, the through space separation is 2.12Å confirming a transition structure with partial bonds.

Typically sp3 bonded carbon has a C-C bondlength of 1.54Å, and sp2 carbon a C=C bondlength of 1.34Å.[1] The observed bond lengths are intermittant between the two, indicative of a transition state with delocalised bonding. In addition, carbon has an atomic van der Waals radius of 1.70Å.[2] thus non-bonded carbons must be 3.4Å apart, the through space separation is 2.12Å confirming a transition structure with partial bonds.

The vibration corresponding to the reaction path at the transition state is shown right - as discussed above - representing synchronous formation of the two bonds. Below this the lowest real vibration is observed, a twisting of the dieneophile fragment over the diene, this is unlikely to correspond to a reaction path, as it does not bring the fragments closer together.

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

If the dieneophile is substituted, with substituents that have π orbitals that can interact with the new double bond that is being formed in the product, then this interaction can stabilise the regiochemistry (i.e. head to tail versus tail to head) of the reaction. In this exercise you will study the nature of the transition structure for the Diels Alder reaction, both for the prototypical reaction and for the case where both diene and dieneophile carry substituents, and where secondary orbital effects are possible. Clearly, the factors that control the nature of the transition state are quantum mechanical in origin and thus we shall use methods based upon quantum chemistry.
Exercise

(iii) To Study the regioselectivity of the Diels Alder Reaction

Cyclohexa-1,3-diene 1 undergoes facile reaction with maleic anhydride 2 to give primarily the endo adduct. The reaction is supposed to be kinetically controlled so that the exo transition state should be higher in energy.

regioslectivity

Locate the transition structures for both 3 and 4. Compare the energies of the endo and exo forms.

Measure the bond lengths of the partly formed σ C-C bonds and the other C-C distances. Make a sketch with the important bond lengths. Measure the orientation, (C-C through space distances between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- for the exo and the “opposite” -CH=CH- for the endo). The structure must be a compromise between steric repulsions of the -CH2-CH2- fragment and the maleic anhydride for the exo versus secondary orbital interactions between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment for the endo.

Plot the HOMO as in the previous exercise. Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”?

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

Rep:Mod:cfl083

2010-11-12T14:22:29Z

Cfl08:

<h1 align="center">Module 3: Charlene Lawton</h1>
<h2>The Cope Rearrangement of 1,5-hexadiene</h2>

The low-energy minima and transition states on the potential energy surface (PES) of 1,5-hexadiene were investigated to determine the preferred reaction mechanism. The [3,3]-sigmatropic shift rearrangement (shown below) occurs in a concerted fashion passing through a chair or boat transition state (TS).

[[Image:Cope_rearrangement_reactncfl08.jpg|centre|The Cope Rearrangement and TS of 1,5-hexadiene|600x203px]]

Optimizing the Reactants and Products

The geometry of 1,5-hexadiene in an anti-periplanar (APP) arrangement and in a gauche arrangement were optimised at the HF/3-21G level of theory. This gave the results tabulated below.

{| class="wikitable" border="1" align="left"
|+ ''Optimised Gauche and APP 1,5-hexadiene conformers''
!!!<jmol><jmolAppletButton><uploadedFileContents>React_anticfl08.mol</uploadedFileContents><text>APP1</text></jmolAppletButton> </jmol>!!<jmol><jmolAppletButton><uploadedFileContents>Gauche hexadienecfl08.mol</uploadedFileContents><text>Gauche</text></jmolAppletButton></jmol>
|-
| Point Group || C2 || C1
|-
| Energy/ Hartree|| -231.69260 || -231.69266
|}
 
It was expected that the gauche linkage would be less stable than the anti linkage; however the computed energies above show the gauche model to be more stable. This is because the additional H-H contacts lead to the gecko effect where the Van der Waals attractive interaction makes the gauche conformation more favourable (ref rzepa). Also there are six possible gauche conformers and only four possible APP conformers meaning the gauche arrangement is entropically favoured. The gauche structure was compared to the reference table and it was confirmed that this conformer is the lowest energy minima. The energy of the APP structure also agrees with the reference energy given in the table. link ref table

Another APP conformer (APP2) of 1,5-hexadiene was optimised at the HF/3-21G level of theory; this gave a structure with Ci symmetry and an energy of -231.69254 Hartree. This structure was then re-optimised to B3LYP/6-31G* level to give a structure with the same point group and an energy of -234.62318 Hartree. The structures look similar - both have a dihedral angle of 180 degrees. The carbon-carbon single bond lengths are slightly shorter whilst the carbon-carbon double bond is slightly longer in the structure optimised to a higher level c.f. the table below.

{| class="wikitable" border="1" align="left"
|+ ''Comparing the optimised APP2 models''
!Bond!!APP2 - HF/3-21G!!APP2 - B3LYP/6-31G*
|-
|C=C||1.32||1.33
|-
|Central C-C || 1.55 || 1.53
|-
|C-C || 1.51 || 1.48
|}
 

<jmol><jmolAppletButton><uploadedFileContents>APP 2cfl08.mol</uploadedFileContents><text>APP2 - HF/3-21G</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>APP 2 opt 2cfl08.mol</uploadedFileContents><text>APP2 - B3LYP/6-31G*</text></jmolAppletButton></jmol>

Vibrational frequencies are the second derivative of the potential energy and therefore minimal energy structures can be confirmed by vibrational analysis; if vibrational frequencies are real and positive this corresponds to a minimum point.

For the higher level calculation: 
Low frequencies: - 38.2274, - 14.7195, 0.0006, 0.0009, 0.0010, 9.6799 
Low frequencies: 73.4693, 74.1196, 119.5657

IR spectrum of the higher level calculation 
[[Image:APP2_IR_speccfl08.jpeg]]

The vibrational analysis computes real properties:
Sum of electronic and zero-point Energies= -233.125690
Sum of electronic and thermal Energies= -233.118189
Sum of electronic and thermal Enthalpies= -233.117245
Sum of electronic and thermal Free Energies= -233.157407

The sum of electronic and zero-point energies is the potential energy at 0 K including the zero-point vibrational energy (E = Eelec + ZPE). The sum of electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans. The sum of electronic and thermal enthalpies contains an additional correction for RT (H = E + RT) which is important when looking at dissociation reactions, and the last sum listed above includes the entropic contribution to the free energy (G = H - TS).

<h2>Optimising the Boat and Chair Transition Structures</h2>

The two possible transition structures (boat or chair) were optimised by 
(1) by computing the force constants at the beginning of the calculation 
(2) using the redundant coordinate editor, and 
(3) using QST2.
The reaction coordinate was visualised, the IRC was run and the activation energies for the Cope rearrangement.

<h3>Chair TS optimisation</h3>

<h4>Method: Computing the force constant matrix</h4>

Initially the delocalised allyl fragment (CH2CHCH2) was optimised (HF/3-21G) and two of these fragments were positioned in the assumed chair transition state with the two terminal carbons approximately 2.2 angstroms apart. The TS was optimised to a TS Berny by computing the force constant matrix (also known as the Hessian) in the first step of the optimization which was then be updated as the optimization proceeded. The opt+freq job type, Hartree Fock method, and the default basis set 3-21G were selected; the force constants were chosen to be calculated once with the additional keywords "Opt=NoEigen" to stop the calculation crashing if more than one imaginary frequency had been discovered. The optimised TS has a distance of 2.02 angstrom between the terminal carbons of each fragment. The frequency analysis showed one imaginary frequency at which corresponds to the Cope rearrangement at -818 cm-1 (0dp).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents>Chair ts guess opt and freqcfl08.mol</uploadedFileContents><text>Chair TS</text></jmolAppletButton></jmol>
 Chair TS Vibration corresponding to the Cope Rearrangement 
[[Image:Chair_TS_cope_vibrationcfl08.gif]]

<h4>Method: Using the redundant coordinate editor </h4>

The TS was then optimised by freezing the reaction coordinate, using Opt=ModRedundant, and minimizing the rest of the molecule. Once the molecule is fully relaxed, the reaction coordinate is unfrozen and the TS optimization is started again. This method means that it may not be necessary to compute the whole Hessian once this has been done, and just differentiating along the reaction coordinate which can save time. Note again the Hartree Fock method and 3-21G basis set were used.
The transition state optimisation was run with the frozen coordinates fixed (distance between the terminal carbons where a bond forms/breaks a bond during the rearrangement fixed to 2.2 angstroms), then repeated with the frozen coordinate bond as a derivative (no force constants calculated). This gave optimised distance between the two fragments as 2.02 angstroms which is in agreement with the TS (Berny) method, shown right; the two methods gave the same result.

<jmol><jmolAppletButton><uploadedFileContents>TS chair freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - fixed bond length</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>TS chair bond freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - optimising bond length</text></jmolAppletButton></jmol>

<h3>Boat TS optimisation</h3>

<h4>Method: The QST2</h4>

The boat TS was optimised using the QST2 method where the reactants and products are specified and the calculation interpolates between the two structures to find the TS between them. The numbering of the carbon and hydrogen atoms were manually changed so that they were numbered in the same way.

Labelled reactant (left) and product (right).

[[Image:Labelled_prod_and_reactantcfl.jpeg]]

The fist QTS2 structure did not give the correct structure - this is because the calculation interpolated linearly between the two structures and translated the top allyl fragment to give a dissociated Chair TS. It was also necessary to manually change the orientation of the reactants and products so that the possibility a rotation around the central bonds was considered. For both the product and reactant, the central C-C-C-C dihedral angle was modified to 0 degrees, then the inside C-C-C bonds were reduced to 100 degrees.

Product and reactants with modified angles.

[[Image:Qst2_modified_angles_inputcfl.jpeg]]

The above input was then optimised to a TS (QST2) using the Hartree Fock method and 3-21G basis set, this gave the TS with C-C distance of 2.14, 2.78 and 2.14 angstrom between the two fragments so that the middle carbons on each fragment are further apart. The optimised TS shows one imaginary frequency at -840 cm-1(0 d.p.).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents> Boat qst2 optcfl08.mol</uploadedFileContents><text>Boat TS</text></jmolAppletButton></jmol>
 Boat TS Vibration corresponding to the Cope Rearrangement 
[[Image:Boat_qst2_opt_cope_vibrationcfl.gif]]

<h4>Method: Intrinsic Reaction Coordinate (IRC)</h4>

The IRC method can be used to predict what conformers of 1,5-hexadiene will develop from the optimised chair and boat transition structures. This method allows the minimum energy path,from a transition structure down to its local minimum on a potential energy surface, to be followed. This creates a series of points by taking small geometry steps in the direction where the gradient or slope of the energy surface is steepest.

Force constants were determined once at the start of the calculation, and the reaction coordinate was ran in just the forward direction (since this is symmetrical), with 50 points. Initially a minimum is not reached; the gradient does not fall to zero. There are three ways a minimum can be reach: 
(1) a normal minimisation can be run on the last point on the IRC, 
(2) the IRC can be restarted with a large number of specified points until a minimum is reached 
(3) the IRC can be re-done with force constants computed at every step.
 Although option three is not always feasible for large systems, it is known to be the most reliable therefore this option was chosen. IRC showed the chair TS to result in the gauche 2 structure (see Image below) with an energy of - 231.69166 au, the paths are illustrated graphically below.
 
Chair TS IRC leads to the Gauche 2 conformer 
[[Image:Irc_gauche_2cfl.jpg]] 
Graph 1: IRC method (force constants computed once) - minimisation not yet reached. 
[[Image:Irc_min_not_reachedcfl.jpeg|centre|IRC - min not reached|542x416px]]
 
Graph 2: IRC method (force constants always computed) - minimisation reached. 
[[Image:Irc_min_reachedcfl.jpg|centre|IRC - min reached|542x416px]] 

<h4>Calculating Activation Energies: re-optimising Boat and Chair TS to higher level of theory</h4>

The activation energies for the reaction via both transition states were calculated by re-optimising the chair and boat transition structures using B3LYP/6-31G* level of theory and carrying out a frequency analysis.
Again one imaginary frequency which connects products to reactants was detected for each structure, showing that the structure is a TS.

Chair TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Chair_TS_reoptcfl08.gif]] 
Boat TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Boat_reopt_507cfl.gif]]

{| class="wikitable" border="1" align="left"
|+ ''Comparing HF/3-21G and B3LYP/6-31G* optimised Boat and Chair TS''
!!!Chair HF/3-21G !!Chair B3LYP/6-31G*!!Boat HF/3-21G !! Boat B3LYP/6-31G*
|-
| Energy/ au|| -231.61932 ||-234.56752||-231.60280 || -234.55445
|-
| C-C distance through space between fragments/angstrom|| 2.02||1.98 ||2.14 ||2.24
|-
| Imaginary Frequency/cm-1 || -818 || -563 ||-840 || -507
|}
 

The geometries are similar but the there is a large energy difference between the two levels of theory, approximately 3 Hartree (~ Kcalmol-1. As a result, usually the potential energy surface is mapped at a low theory level and then re-optimised at a higher level to provide more efficient calculations.

The activation energies at 0 Kelvin for the cope rearrangement were calculated by taking the difference between the energy of the reactant and the energy of the transition state.

{| class="wikitable" border="1" align="centre"
|+ ''Activation Energies at 0K (kcal/mol)''
! !! HF/3-21G !! B3LYP/6-31G* !! Experiment
|-
| Chair||45.4 || 35.0 || 33±0.5
|-
|Boat||58.1 || 43.1 || 44.7±2.0
|}

As the calculation is down to a higher level of theory, the activation energy values approach the experimental values. In order to completely match the experimental data an even higher level of theory would have to be used- the observed difference is due to the approximations made in the calculation.

<h1>The Diels Alder Cycloaddition</h1>

<h2>Exercise 1: MOs of Reactants</h2>
Cis-butadiene and ethylene were optimised using the semi-empirical method (AM1).
<jmol><jmolAppletButton><uploadedFileContents>BUTADIENE_OPTcfl08.mol</uploadedFileContents><text>Cis-butadiene</text></jmolAppletButton></jmol>
<jmol><jmolAppletButton><uploadedFileContents>ETHYLENE_OPTcfl08.mol</uploadedFileContents><text>Ethylene</text></jmolAppletButton> </jmol>
 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Reactants''
! Cis-Butadiene LUMO !! Cis-butadiene HOMO !! Ethylene LUMO !! Ethylene HOMO
|-
| [[Image:Cis_butadiene_LUMO.cfljpg]] || [[Image:Cis_butadiene_HOMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_LUMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_HOMOcfl.jpg|centre|HOMO B|244x190px]]
|-
| symmetric || antisymmetric || antisymmetric|| symmetric
|}
 

'''Discussion of MOs'''

The [4s + 2s] pericyclic reaction between ethylene and cis-butadiene depends on the HOMO/LUMO interactions of the molecules. The concerted reaction is allowed if the HOMO of one reactant can interact with the LUMO of the other reactant and when the HOMO-LUMO interaction has significant overlap density. There is no overlap density if the orbitals have different symmetry properties.
The HOMO of ethylene and the LUMO of butadiene are both symmetric with respect to the reflection plane and the LUMO of ethylene and the HOMO of butadiene are both antisymmetric. Therefore, the HOMO-LUMO pairs can interact and the principal orbital interactions involve the π/ π* orbitals of ethylene and the HOMO/LUMO of butadiene; π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene.

<h2>Exercise 1: Computation of the Transition State geometry</h2>

The transition state was optimised using the redundant coordinate editor (Opt=ModRedundant) - the reaction coordinate was frozen and the rest of the molecule minimised (distance between ethylene and butadiene set to 2.2 angstrom) then reaction coordinate was unfrozen and the structure minimised to a TS(Berny). The semi-empirical AM1 method was used.
<jmol><jmolAppletButton><uploadedFileContents>But_eth_TScfl08.mol</uploadedFileContents><text>Optimised Transition State</text></jmolAppletButton></jmol>

 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Optimised Transition State''
!HOMO-1!!HOMO !!LUMO !! LUMO-1
|-
| [[Image:But_eth_TS_HOMO-1cfl.jpeg]] || [[Image:But_eth_TS_HOMO_scfl.jpeg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO_scfl.jpg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO+1cfl.jpeg|centre|HOMO B|224x263px]]
|-
| symmetric || antisymmetric || symmetric|| antisymmetric
|}
 

'''Discussion of TS Geometry and MOs'''

The antisymmetric HOMO of the transition structure, shown above, is formed from the HOMO of cis-butadiene interacting with the LUMO of ethylene. This reaction is allowed beaucse the two interacting orbitals have the same symmetry and significant overlap density.

Below is the optimsied TS with the bond lengths labelled 
[[Image:Diels_Alder_TS_bondlengthscfl08.jpg]]

- the structure was confirmed to be a TS by frequency analysis; the frequency analysis showed one imaginary frequency at -956cm-1.

'''Vibrational Analysis of TS'''

The structure was confirmed to be a TS by frequency analysis
 Transition State Imaginary Frequency [[Image:Diels_alder_TS_b2_freqcfl.gif‎‎]]

Notably the LUMO of the transition state is formed by the combination of the ethylene HOMO and cis-butadiene LUMO (both symmetric, s); the ethylene HOMO and LUMO are of course the pi and pi* bonding and anti-bonding MOs.
Which MOs of butadiene and ethylene have been used to form this MO? Explain why the reaction is allowed.
The π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene and the resulting HOMO is asymmetric.

What are typical sp3 and sp2 C-C bondlengths? What is the van der Waals radius of the C atom? What can you conclude about the C-C bondlength of the partly formed σ C-C bonds in the TS.

ypically sp3 bonded carbon has a C-C bondlength of 1.54Å, and sp2 carbon a C=C bondlength of 1.34Å.[1] The observed bond lengths are intermittant between the two, indicative of a transition state with delocalised bonding. In addition, carbon has an atomic van der Waals radius of 1.70Å.[2] thus non-bonded carbons must be 3.4Å apart, the through space separation is 2.12Å confirming a transition structure with partial bonds.

The vibration corresponding to the reaction path at the transition state is shown right - as discussed above - representing synchronous formation of the two bonds. Below this the lowest real vibration is observed, a twisting of the dieneophile fragment over the diene, this is unlikely to correspond to a reaction path, as it does not bring the fragments closer together.

. Is the formation of the two bonds synchronous or asynchronous? How does this compare with the lowest positive frequency?
Typically sp3 bonded carbon has a C-C bondlength of 1.54Å, and sp2 carbon a C=C bondlength of 1.34Å.[1] The observed bond lengths are intermittant between the two, indicative of a transition state with delocalised bonding. In addition, carbon has an atomic van der Waals radius of 1.70Å.[2] thus non-bonded carbons must be 3.4Å apart, the through space separation is 2.12Å confirming a transition structure with partial bonds.

The vibration corresponding to the reaction path at the transition state is shown right - as discussed above - representing synchronous formation of the two bonds. Below this the lowest real vibration is observed, a twisting of the dieneophile fragment over the diene, this is unlikely to correspond to a reaction path, as it does not bring the fragments closer together.

The HOMO of the transition structure is shown above, being anti-symmetric (a); the HOMO of cis-butadiene interacts with the LUMO of ethylene to form the transition state HOMO, as shown below.

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

If the dieneophile is substituted, with substituents that have π orbitals that can interact with the new double bond that is being formed in the product, then this interaction can stabilise the regiochemistry (i.e. head to tail versus tail to head) of the reaction. In this exercise you will study the nature of the transition structure for the Diels Alder reaction, both for the prototypical reaction and for the case where both diene and dieneophile carry substituents, and where secondary orbital effects are possible. Clearly, the factors that control the nature of the transition state are quantum mechanical in origin and thus we shall use methods based upon quantum chemistry.
Exercise

(iii) To Study the regioselectivity of the Diels Alder Reaction

Cyclohexa-1,3-diene 1 undergoes facile reaction with maleic anhydride 2 to give primarily the endo adduct. The reaction is supposed to be kinetically controlled so that the exo transition state should be higher in energy.

regioslectivity

Locate the transition structures for both 3 and 4. Compare the energies of the endo and exo forms.

Measure the bond lengths of the partly formed σ C-C bonds and the other C-C distances. Make a sketch with the important bond lengths. Measure the orientation, (C-C through space distances between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- for the exo and the “opposite” -CH=CH- for the endo). The structure must be a compromise between steric repulsions of the -CH2-CH2- fragment and the maleic anhydride for the exo versus secondary orbital interactions between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment for the endo.

Plot the HOMO as in the previous exercise. Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”?

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

File:Diels Alder TS bondlengthscfl08.jpg

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2010-11-11T19:18:52Z

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Rep:Mod:cfl083

2010-11-11T19:13:49Z

Cfl08:

<h1 align="center">Module 3: Charlene Lawton</h1>
<h2>The Cope Rearrangement of 1,5-hexadiene</h2>

The low-energy minima and transition states on the potential energy surface (PES) of 1,5-hexadiene were investigated to determine the preferred reaction mechanism. The [3,3]-sigmatropic shift rearrangement (shown below) occurs in a concerted fashion passing through a chair or boat transition state (TS).

[[Image:Cope_rearrangement_reactncfl08.jpg|centre|The Cope Rearrangement and TS of 1,5-hexadiene|600x203px]]

Optimizing the Reactants and Products

The geometry of 1,5-hexadiene in an anti-periplanar (APP) arrangement and in a gauche arrangement were optimised at the HF/3-21G level of theory. This gave the results tabulated below.

{| class="wikitable" border="1" align="left"
|+ ''Optimised Gauche and APP 1,5-hexadiene conformers''
!!!<jmol><jmolAppletButton><uploadedFileContents>React_anticfl08.mol</uploadedFileContents><text>APP1</text></jmolAppletButton> </jmol>!!<jmol><jmolAppletButton><uploadedFileContents>Gauche hexadienecfl08.mol</uploadedFileContents><text>Gauche</text></jmolAppletButton></jmol>
|-
| Point Group || C2 || C1
|-
| Energy/ Hartree|| -231.69260 || -231.69266
|}
 
It was expected that the gauche linkage would be less stable than the anti linkage; however the computed energies above show the gauche model to be more stable. This is because the additional H-H contacts lead to the gecko effect where the Van der Waals attractive interaction makes the gauche conformation more favourable (ref rzepa). Also there are six possible gauche conformers and only four possible APP conformers meaning the gauche arrangement is entropically favoured. The gauche structure was compared to the reference table and it was confirmed that this conformer is the lowest energy minima. The energy of the APP structure also agrees with the reference energy given in the table. link ref table

Another APP conformer (APP2) of 1,5-hexadiene was optimised at the HF/3-21G level of theory; this gave a structure with Ci symmetry and an energy of -231.69254 Hartree. This structure was then re-optimised to B3LYP/6-31G* level to give a structure with the same point group and an energy of -234.62318 Hartree. The structures look similar - both have a dihedral angle of 180 degrees. The carbon-carbon single bond lengths are slightly shorter whilst the carbon-carbon double bond is slightly longer in the structure optimised to a higher level c.f. the table below.

{| class="wikitable" border="1" align="left"
|+ ''Comparing the optimised APP2 models''
!Bond!!APP2 - HF/3-21G!!APP2 - B3LYP/6-31G*
|-
|C=C||1.32||1.33
|-
|Central C-C || 1.55 || 1.53
|-
|C-C || 1.51 || 1.48
|}
 

<jmol><jmolAppletButton><uploadedFileContents>APP 2cfl08.mol</uploadedFileContents><text>APP2 - HF/3-21G</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>APP 2 opt 2cfl08.mol</uploadedFileContents><text>APP2 - B3LYP/6-31G*</text></jmolAppletButton></jmol>

Vibrational frequencies are the second derivative of the potential energy and therefore minimal energy structures can be confirmed by vibrational analysis; if vibrational frequencies are real and positive this corresponds to a minimum point.

For the higher level calculation: 
Low frequencies: - 38.2274, - 14.7195, 0.0006, 0.0009, 0.0010, 9.6799 
Low frequencies: 73.4693, 74.1196, 119.5657

IR spectrum of the higher level calculation 
[[Image:APP2_IR_speccfl08.jpeg]]

The vibrational analysis computes real properties:
Sum of electronic and zero-point Energies= -233.125690
Sum of electronic and thermal Energies= -233.118189
Sum of electronic and thermal Enthalpies= -233.117245
Sum of electronic and thermal Free Energies= -233.157407

The sum of electronic and zero-point energies is the potential energy at 0 K including the zero-point vibrational energy (E = Eelec + ZPE). The sum of electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans. The sum of electronic and thermal enthalpies contains an additional correction for RT (H = E + RT) which is important when looking at dissociation reactions, and the last sum listed above includes the entropic contribution to the free energy (G = H - TS).

<h2>Optimising the Boat and Chair Transition Structures</h2>

The two possible transition structures (boat or chair) were optimised by 
(1) by computing the force constants at the beginning of the calculation 
(2) using the redundant coordinate editor, and 
(3) using QST2.
The reaction coordinate was visualised, the IRC was run and the activation energies for the Cope rearrangement.

<h3>Chair TS optimisation</h3>

<h4>Method: Computing the force constant matrix</h4>

Initially the delocalised allyl fragment (CH2CHCH2) was optimised (HF/3-21G) and two of these fragments were positioned in the assumed chair transition state with the two terminal carbons approximately 2.2 angstroms apart. The TS was optimised to a TS Berny by computing the force constant matrix (also known as the Hessian) in the first step of the optimization which was then be updated as the optimization proceeded. The opt+freq job type, Hartree Fock method, and the default basis set 3-21G were selected; the force constants were chosen to be calculated once with the additional keywords "Opt=NoEigen" to stop the calculation crashing if more than one imaginary frequency had been discovered. The optimised TS has a distance of 2.02 angstrom between the terminal carbons of each fragment. The frequency analysis showed one imaginary frequency at which corresponds to the Cope rearrangement at -818 cm-1 (0dp).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents>Chair ts guess opt and freqcfl08.mol</uploadedFileContents><text>Chair TS</text></jmolAppletButton></jmol>
 Chair TS Vibration corresponding to the Cope Rearrangement 
[[Image:Chair_TS_cope_vibrationcfl08.gif]]

<h4>Method: Using the redundant coordinate editor </h4>

The TS was then optimised by freezing the reaction coordinate, using Opt=ModRedundant, and minimizing the rest of the molecule. Once the molecule is fully relaxed, the reaction coordinate is unfrozen and the TS optimization is started again. This method means that it may not be necessary to compute the whole Hessian once this has been done, and just differentiating along the reaction coordinate which can save time. Note again the Hartree Fock method and 3-21G basis set were used.
The transition state optimisation was run with the frozen coordinates fixed (distance between the terminal carbons where a bond forms/breaks a bond during the rearrangement fixed to 2.2 angstroms), then repeated with the frozen coordinate bond as a derivative (no force constants calculated). This gave optimised distance between the two fragments as 2.02 angstroms which is in agreement with the TS (Berny) method, shown right; the two methods gave the same result.

<jmol><jmolAppletButton><uploadedFileContents>TS chair freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - fixed bond length</text></jmolAppletButton></jmol><jmol><jmolAppletButton><uploadedFileContents>TS chair bond freeze optcfl08.mol</uploadedFileContents><text>Chair TS opt - optimising bond length</text></jmolAppletButton></jmol>

<h3>Boat TS optimisation</h3>

<h4>Method: The QST2</h4>

The boat TS was optimised using the QST2 method where the reactants and products are specified and the calculation interpolates between the two structures to find the TS between them. The numbering of the carbon and hydrogen atoms were manually changed so that they were numbered in the same way.

Labelled reactant (left) and product (right).

[[Image:Labelled_prod_and_reactantcfl.jpeg]]

The fist QTS2 structure did not give the correct structure - this is because the calculation interpolated linearly between the two structures and translated the top allyl fragment to give a dissociated Chair TS. It was also necessary to manually change the orientation of the reactants and products so that the possibility a rotation around the central bonds was considered. For both the product and reactant, the central C-C-C-C dihedral angle was modified to 0 degrees, then the inside C-C-C bonds were reduced to 100 degrees.

Product and reactants with modified angles.

[[Image:Qst2_modified_angles_inputcfl.jpeg]]

The above input was then optimised to a TS (QST2) using the Hartree Fock method and 3-21G basis set, this gave the TS with C-C distance of 2.14, 2.78 and 2.14 angstrom between the two fragments so that the middle carbons on each fragment are further apart. The optimised TS shows one imaginary frequency at -840 cm-1(0 d.p.).

Optimised Chair TS 
<jmol><jmolAppletButton><uploadedFileContents> Boat qst2 optcfl08.mol</uploadedFileContents><text>Boat TS</text></jmolAppletButton></jmol>
 Boat TS Vibration corresponding to the Cope Rearrangement 
[[Image:Boat_qst2_opt_cope_vibrationcfl.gif]]

<h4>Method: Intrinsic Reaction Coordinate (IRC)</h4>

The IRC method can be used to predict what conformers of 1,5-hexadiene will develop from the optimised chair and boat transition structures. This method allows the minimum energy path,from a transition structure down to its local minimum on a potential energy surface, to be followed. This creates a series of points by taking small geometry steps in the direction where the gradient or slope of the energy surface is steepest.

Force constants were determined once at the start of the calculation, and the reaction coordinate was ran in just the forward direction (since this is symmetrical), with 50 points. Initially a minimum is not reached; the gradient does not fall to zero. There are three ways a minimum can be reach: 
(1) a normal minimisation can be run on the last point on the IRC, 
(2) the IRC can be restarted with a large number of specified points until a minimum is reached 
(3) the IRC can be re-done with force constants computed at every step.
 Although option three is not always feasible for large systems, it is known to be the most reliable therefore this option was chosen. IRC showed the chair TS to result in the gauche 2 structure (see Image below) with an energy of - 231.69166 au, the paths are illustrated graphically below.
 
Chair TS IRC leads to the Gauche 2 conformer 
[[Image:Irc_gauche_2cfl.jpg]] 
Graph 1: IRC method (force constants computed once) - minimisation not yet reached. 
[[Image:Irc_min_not_reachedcfl.jpeg|centre|IRC - min not reached|542x416px]]
 
Graph 2: IRC method (force constants always computed) - minimisation reached. 
[[Image:Irc_min_reachedcfl.jpg|centre|IRC - min reached|542x416px]] 

<h4>Calculating Activation Energies: re-optimising Boat and Chair TS to higher level of theory</h4>

The activation energies for the reaction via both transition states were calculated by re-optimising the chair and boat transition structures using B3LYP/6-31G* level of theory and carrying out a frequency analysis.
Again one imaginary frequency which connects products to reactants was detected for each structure, showing that the structure is a TS.

Chair TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Chair_TS_reoptcfl08.gif]] 
Boat TS Imaginary Frequency (method:B3LYP/6-31G*) 
[[Image:Boat_reopt_507cfl.gif]]

{| class="wikitable" border="1" align="left"
|+ ''Comparing HF/3-21G and B3LYP/6-31G* optimised Boat and Chair TS''
!!!Chair HF/3-21G !!Chair B3LYP/6-31G*!!Boat HF/3-21G !! Boat B3LYP/6-31G*
|-
| Energy/ au|| -231.61932 ||-234.56752||-231.60280 || -234.55445
|-
| C-C distance through space between fragments/angstrom|| 2.02||1.98 ||2.14 ||2.24
|-
| Imaginary Frequency/cm-1 || -818 || -563 ||-840 || -507
|}
 

The geometries are similar but the there is a large energy difference between the two levels of theory, approximately 3 Hartree (~ Kcalmol-1. As a result, usually the potential energy surface is mapped at a low theory level and then re-optimised at a higher level to provide more efficient calculations.

The activation energies at 0 Kelvin for the cope rearrangement were calculated by taking the difference between the energy of the reactant and the energy of the transition state.

{| class="wikitable" border="1" align="centre"
|+ ''Activation Energies at 0K (kcal/mol)''
! !! HF/3-21G !! B3LYP/6-31G* !! Experiment
|-
| Chair||45.4 || 35.0 || 33±0.5
|-
|Boat||58.1 || 43.1 || 44.7±2.0
|}

As the calculation is down to a higher level of theory, the activation energy values approach the experimental values. In order to completely match the experimental data an even higher level of theory would have to be used- the observed difference is due to the approximations made in the calculation.

<h1>The Diels Alder Cycloaddition</h1>

<h2>Exercise 1: MOs of Reactants</h2>
Cis-butadiene and ethylene were optimised using the semi-empirical method (AM1).
<jmol><jmolAppletButton><uploadedFileContents>BUTADIENE_OPTcfl08.mol</uploadedFileContents><text>Cis-butadiene</text></jmolAppletButton></jmol>
<jmol><jmolAppletButton><uploadedFileContents>ETHYLENE_OPTcfl08.mol</uploadedFileContents><text>Ethylene</text></jmolAppletButton> </jmol>
 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Reactants''
! Cis-Butadiene LUMO !! Cis-butadiene HOMO !! Ethylene LUMO !! Ethylene HOMO
|-
| [[Image:Cis_butadiene_LUMO.cfljpg]] || [[Image:Cis_butadiene_HOMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_LUMOcfl.jpg|centre|HOMO B|244x190px]]|| [[Image:Ethylene_HOMOcfl.jpg|centre|HOMO B|244x190px]]
|-
| symmetric || antisymmetric || antisymmetric|| symmetric
|}
 

'''Discussion of MOs'''

The pericyclic reaction between ethylene and cis-butadiene depends on the HOMO/LUMO interactions of the molecules. The concerted reaction is allowed if the HOMO of one reactant can interact with the LUMO of the other reactant and when the HOMO-LUMO interaction has significant overlap density. There is no overlap density if the orbitals have different symmetry properties.

The HOMO of ethylene and the LUMO of butadiene are both symmetric with respect to the reflection plane and the LUMO of ethylene and the HOMO of butadiene are both antisymmetric. Therefore, the HOMO-LUMO pairs can interact and the principal orbital interactions involve the π/ π* orbitals of ethylene and the HOMO/LUMO of butadiene. This is referred to as [4s + 2s] cycloaddition.

<h2>Exercise 1: Computation of the Transition State geometry</h2>

The transition state was optimised using the redundant coordinate editor (Opt=ModRedundant) - the reaction coordinate was frozen and the rest of the molecule minimised (distance between ethylene and butadiene set to 2.2 angstrom) then reaction coordinate was unfrozen and the structure minimised to a TS(Berny). The semi-empirical AM1 method was used - the structure was confirmed to be a TS by frequency analysis; the frequency analysis showed one imaginary frequency at -956cm-1
<jmol><jmolAppletButton><uploadedFileContents></uploadedFileContents><text>Optimised Transition State</text></jmolAppletButton></jmol>
<jmol><jmolAppletButton><uploadedFileContents></uploadedFileContents><text>TS Imaginary Frequency</text></jmolAppletButton></jmol>

 
{| class="wikitable" border="1" align="left"
|+ ''MOs of Optimised Transition State''
!HOMO-1!!HOMO !!LUMO !! LUMO-1
|-
| [[Image:But_eth_TS_HOMO-1cfl.jpeg]] || [[Image:But_eth_TS_HOMO_scfl.jpeg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO_scfl.jpg|centre|HOMO B|224x263px]]|| [[Image:But_eth_TS_LUMO+1cfl.jpeg|centre|HOMO B|224x263px]]
|-
| symmetric || antisymmetric || symmetric|| antisymmetric
|}
 

The π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene adn the resulting HOMO is assymmetric.

If the dieneophile is substituted, with substituents that have π orbitals that can interact with the new double bond that is being formed in the product, then this interaction can stabilise the regiochemistry (i.e. head to tail versus tail to head) of the reaction. In this exercise you will study the nature of the transition structure for the Diels Alder reaction, both for the prototypical reaction and for the case where both diene and dieneophile carry substituents, and where secondary orbital effects are possible. Clearly, the factors that control the nature of the transition state are quantum mechanical in origin and thus we shall use methods based upon quantum chemistry.
Exercise

Use the the AM1 semi-empirical molecular orbital method for these calculations (to start with).

Confirm you have obtained a transition structure for the Diels Alder reaction!

guessing the transition structure

Once you have obtained the correct structure, plot the HOMO as in (i). Rotate the molecule so that the symmetry and nodal properties of the system can be interpreted, and save a copy of the image.

(iii) To Study the regioselectivity of the Diels Alder Reaction

Cyclohexa-1,3-diene 1 undergoes facile reaction with maleic anhydride 2 to give primarily the endo adduct. The reaction is supposed to be kinetically controlled so that the exo transition state should be higher in energy.

regioslectivity

Locate the transition structures for both 3 and 4. Compare the energies of the endo and exo forms.

Measure the bond lengths of the partly formed σ C-C bonds and the other C-C distances. Make a sketch with the important bond lengths. Measure the orientation, (C-C through space distances between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- for the exo and the “opposite” -CH=CH- for the endo). The structure must be a compromise between steric repulsions of the -CH2-CH2- fragment and the maleic anhydride for the exo versus secondary orbital interactions between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment for the endo.

Plot the HOMO as in the previous exercise. Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”?

Suggested Discussion

Use this template as a guide. Screen images can be saved from the GaussView File menu.

For cis butadiene:
Plot the HOMO and LUMO and determine the symmetry (symmetric or anti-symmetric) with respect to the plane.

For the ethylene+cis butadiene transition structure:
Sketch HOMO and LUMO, labeling each as symmetric or anti symmetric.

Show the geometry of the transition structure, including the bond-lengths of the partly formed σ C-C bonds.

What are typical sp3 and sp2 C-C bondlengths? What is the van der Waals radius of the C atom? What can you conclude about the C-C bondlength of the partly formed σ C-C bonds in the TS.

Illustrate the vibration that corresponds to the reaction path at the transition state. Is the formation of the two bonds synchronous or asynchronous? How does this compare with the lowest positive frequency?

Is the HOMO at the transition structure s or a?

Which MOs of butadiene and ethylene have been used to form this MO? Explain why the reaction is allowed.

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

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?