ChemWiki - User contributions [en]

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:47:14Z

Lh2213: /* Associated Output Files */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

<jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

<jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|B3LYP/631G* optimised boat transition structure
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|B3LYP/631G* optimised chair transition structure
|-
|[[File:OTRAP(A)-FREQ.LOG]]
|HF frequency analysis of the 1,5 Antiperiplanar conformer of 1,5 heaxadiene
|-
|[[File:OTRAP(B)-FREQ.LOG]]
|HF frequency analysis of the 1,5 Gauche conformer of 1,5 heaxadiene
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

<jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO transition structures were assessed using maleic annhydride and cyclohexa-1,3-diene. In the ENDO structure (and not in the EXO structure), an orbital interaction between the -(CO)-O-(CO)- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 Hartrees respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively is quite small, and it would be expected that thermal influences could nullify this influence. A steric clash with the hydrogen atoms of the <math>-CH_2-CH_2-</math> groups above the -(CO)-O-(CO)- system may also have destabilized the EXO structure. However this interaction would have most likely been overruled by the secondary orbital overlap stabilization of the ENDO structure.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

<jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

File:OTRAP(B)-FREQ.LOG

2016-02-26T11:44:46Z

Lh2213:

File:OTRAP(A)-FREQ.LOG

2016-02-26T11:43:48Z

Lh2213:

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:42:32Z

Lh2213: /* Observing the Regioselectivity of the Diels-Alder Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

<jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

<jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

<jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO transition structures were assessed using maleic annhydride and cyclohexa-1,3-diene. In the ENDO structure (and not in the EXO structure), an orbital interaction between the -(CO)-O-(CO)- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 Hartrees respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively is quite small, and it would be expected that thermal influences could nullify this influence. A steric clash with the hydrogen atoms of the <math>-CH_2-CH_2-</math> groups above the -(CO)-O-(CO)- system may also have destabilized the EXO structure. However this interaction would have most likely been overruled by the secondary orbital overlap stabilization of the ENDO structure.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

<jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:35:21Z

Lh2213: /* Observing the Regioselectivity of the Diels-Alder Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

<jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

<jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

<jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO transition structures were assessed using maleic annhydride and cyclohexa-1,3-diene. In the ENDO structure (and not in the EXO structure), an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

<jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:34:19Z

Lh2213: /* Observing the Regioselectivity of the Diels-Alder Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

<jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

<jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

<jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3-diene. In the Endo product (and not in the exo-product), an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

<jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:31:07Z

Lh2213:

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

<jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

<jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

<jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

<jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

<jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:29:20Z

Lh2213: /* References */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. (accessed 22/02/2016)
#Mudar A. Abdulsattar and Khalil H. Al-Bayati. Corrections and parametrization of semiempirical large unit cell method for covalent semiconductors. ''Phys. Rev. B ''2007; 75(245201): 1-8. DOI:http://dx.doi.org/10.1103/PhysRevB.75.245201 (accessed 22/02/2016).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412 (accessed 22/02/2016)

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:22:54Z

Lh2213: /* References */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data. From this, it can be seen that the Chair conformer is preferred due to the lower activation energy. Comparison of the calculation methods shows that there are significant differences in the energies, leading to variable accuracy depending on which method is used. In addition, the energy trends between calculation type do not remain constant. This makes experimental comparison less facile, in terms of accuracy, between calculation types.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, stabilized by a secondary orbital interaction and proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' 25 (12): 1463–1473. doi:10.1002/jcc.20078. PMID 15224390.
#Mudar A. Abdulsattar and Khalil H. Al-Bayati, ‘Corrections and parameterization of semiempirical large unit cell method for covalent semiconductors’, Phys. Rev. B 75, 245201 (2007).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:09:45Z

Lh2213: /* References */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, proceeding via a synchronous molecular vibration within the transition state.

== References ==
#Charlotte Froese Fischer. General Hartree-Fock program. Computer Physics Communications 1987; 43(3): 355-365. doi:10.1016/0010-4655(87)90053-1 (accessed 22/02/2016).http://www.sciencedirect.com/science/article/pii/0010465587900531
#Motohiro Nishio. CH/p hydrogen bonds in crystals. CrystEngComm 2004; 6(27): 130-158. DOI: 10.1039/b313104a (accessed 22/02/2016).http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' '''25''' (12): 1463–1473. doi:10.1002/jcc.20078. PMID 15224390.
#Mudar A. Abdulsattar and Khalil H. Al-Bayati, ‘Corrections and parameterization of semiempirical large unit cell method for covalent semiconductors’, Phys. Rev. B 75, 245201 (2007).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412

Rep:Mod:L Harnett Trans Struct

2016-02-26T11:01:29Z

Lh2213: /* References */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.<sub>[1]</sub>

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).<sub>[3]</sub>

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.<sub>[4]</sub>

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction.<sub>[2,5]</sub> However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, proceeding via a synchronous molecular vibration within the transition state.

== References ==
#<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>
#http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a.
#Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". ''Journal of Computational Chemistry'' '''25''' (12): 1463–1473. doi:10.1002/jcc.20078. PMID 15224390.
#Mudar A. Abdulsattar and Khalil H. Al-Bayati, ‘Corrections and parameterization of semiempirical large unit cell method for covalent semiconductors’, Phys. Rev. B 75, 245201 (2007).
#BRANDON G. ROCQUE , JASON M. GONZALES & HENRY F. SCHAEFER III(2002) An analysis of the conformers of 1,5-hexadiene, Molecular Physics, 100:4, 441-446, DOI:10.1080/00268970110081412

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:49:29Z

Lh2213: /* Associated Output Files */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

== Conclusion ==
To conclude, the lowest energy conformer of 1,5-Hexadiene was found to be the "Gauche 3" conformer resulting from a π-π interaction between the terminal C=C Groups. The possible transition structures (Chair and Boat) of the Cope Rearrangement were calculated via the Berny, Frozen Coordinate and QST2 methods. It was found that the chair conformer was lower in energy and therefore the preferred transition structure. The Cope rearrangement product conformer was calculated, via IRC calculations, as the "Gauche 2" conformer. The Diels-Alder Cycloaddition was then simulated using the Berny Method. The ENDO product was found to be both the thermodynamic and kinetic product, proceeding via a synchronous molecular vibration within the transition state.

== References ==
#

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:36:11Z

Lh2213: /* Associated Output Files */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

The reactive vibrations of this system remained largely unchanged compared to the cis-butadiene/ethylene example. A lowering of the ENDO vibration frequency further supports its position as the preferred product due to the the direct correlation between frequency and energy.

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO). Notice how the values for r and r' are smaller in the ENDO product. This further supports the concept of a secondary orbital pulling the reagents together. Thus stabilizing the transition structure, lowering the energy and shortening these distances.

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:26:43Z

Lh2213: /* Finding the Transition State Geometry of the Diels-Alder Prototype Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in asynchronously and not in the plane of the reacting bonds, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion and have the wrong relative phases.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:24:38Z

Lh2213: /* Finding the Transition State Geometry of the Diels-Alder Prototype Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

<jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:23:13Z

Lh2213: /* The ENDO Transition State */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assessed using maleic annhydride and cyclohexa-1,3 diene. In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π system of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated as -0.16017080 Hartrees and -0.15990915 respectively. The lower ENDO product energy indicates that it may also be the thermodynamic product of the reaction. However, the difference in energy relatively quite small, and it would be expected that thermal influences could nullify this influence.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T10:15:12Z

Lh2213: /* The ENDO Transition State */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
In the Endo product, an orbital interaction between the -(CO)-O-(CO- π system of the dieneophile and C=C-C=C π sytem of the diene can be clearly visualized (SEE: Fig 52-55). This secondary orbital overlap stabilizes the transition structure. Lowering its energy, relative to the EXO transition structure, and thus explaining why the ENDO product is the kinetic product (due to the lowered relative activation energy). In addition, the energies of the ENDO and EXO products were calculated<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: EXO Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: ENDO Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:54:32Z

Lh2213: /* Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some small variations. This indicates that the the Cope rearrangement does not relax back down to the "Gauche 3" conformer immediately. It must form the"Gauche 2" conformer and then rotate accordingly. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30'''E= -231.69157910. Initial IRC calculation (50 points)
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31''' E=-231.69166702 (IRC then subsequent HF minimisation)
File:OTCABTS(F)II.png|'''Fig 32''' E=-231.69157910 (IRC 110 point calculation)
File:OTCABTS(F)III.png|'''Fig 33''' E=-231.68295296 (IRC 50 points with force constants plotted at every step)
</gallery>

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:50:30Z

Lh2213: /* Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====
Starting from the HF optimized the Intrinsic Reaction Coordinate method (IRC) was used to find the product conformer of the Cope Rearrangement. The method works by following the slope of the potential energy surface until it finds a minimum. First an IRC calculation of 50 points was used to find the product conformer. This gave a structure that closely resembled the "Gauche 2" conformer (Energy=-231.69167, Point Group C2). Further calculations were done to ensure that this was the resulting conformer, all giving the same approximate result with some variation. <gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:42:35Z

Lh2213: /* The HF Calculation Method */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation, which often leads to deviations from experimental values.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====
The DFT method of calculation uses functionals (function of other functions, in this case, the function is electron density) to calculate the molecular wave function. An advantage of DFT of HF is that it considers electron correlation as well as the exchange interactions between electrons(when electrons are found closer together that expected, leading to a lowering of the wave-function expectation values).

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====
Semi-Empirical methods are based on HF calculations but utilize several additional approximations in the calculations as well as using experimentally derived parameters. Austin Model 1 is an example of such a calculation. Such methods are normally used when the HF calculation is too computationally demanding. In addition the experimentally derived parameters add some electron correlation considerations, which aid in the the accuracy of the calculation.

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.
* * is a polarization function that gives the basis set more flexibility. Allowing it to represent asymmetric systems more closely.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:32:31Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|<nowiki>-231.466701</nowiki>
|<nowiki>-231.461341</nowiki>
|0.072838
|0.071224
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|0.05429
|0.05286
|-
|Boat HF/3-21G
|<nowiki>-231.539486</nowiki>
|<nowiki>-231.532646</nowiki>
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|0.088558
|0.087347
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|0.066877
|0.065861
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:23:34Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|
|
|-
|Boat HF/3-21G
|231.539486
|231.532646
|<nowiki>-231.450928</nowiki>
|<nowiki>-231.445299</nowiki>
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:21:15Z

Lh2213: /* The HF Calculation Method */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|<nowiki>-231.539539</nowiki>
|<nowiki>-231.532565</nowiki>
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|
|
|-
|Boat HF/3-21G
|231.539486
|231.532646
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:09:15Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.414929</nowiki>
|<nowiki>-234.409009</nowiki>
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|<nowiki>-234.402342</nowiki>
|<nowiki>-234.396008</nowiki>
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:03:38Z

Lh2213: /* Associated Output Files */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|-
|[[File:OTCABTS(G)-BOAT.LOG]]
|
|-
|[[File:OTCABTS(G)-CHAIR.LOG]]
|
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

File:OTCABTS(G)-CHAIR.LOG

2016-02-26T08:03:23Z

Lh2213:

Rep:Mod:L Harnett Trans Struct

2016-02-26T08:02:08Z

Lh2213: /* The Basis Sets */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system. An important consideration is that the HF method does not consider electron correlation

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
The 3-21G and 6-31G* basis sets are used for the HF and DFT calculations respectively. Both of these are examples of Split Valence basis sets of the form: X-YZG.
* X specifies is the number of primitive Gaussian functions used in each atomic orbital basis function.
* Y and Z each represent primitive Gaussian functions that are linearly combined to represent the valence orbitals.

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

File:OTCABTS(G)-BOAT.LOG

2016-02-26T08:01:32Z

Lh2213:

Rep:Mod:L Harnett Trans Struct

2016-02-26T01:07:30Z

Lh2213: /* Finding the Transition State Geometry of the Diels-Alder Prototype Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

In contrast, the lowest, non-imaginary vibration involves the two components oscillating in different planes, indicating why this is a non-reactive vibration. The components simply do not interact in the same plane of motion.

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T01:02:36Z

Lh2213: /* Observing the Regioselectivity of the Diels-Alder Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
Starting from the product and partially breaking the two reactive C-C bonds, an initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. The visualized MOs corresponded directly to the expected results in that the diene LUMO overlapped in phase with the dieneophile HOMO, and the dieneophile LUMO overlapped in phase with the diene HOMO. This was deduced via comparison with the reagent MOs and symmetries in the previous section. Note also that the Diels Alder reaction is a thermally allowed [4+2]-cycloaddition via Woodward-Hoffman analysis, which also explains its reactivity.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

The vibrations of the transition structure were also simulated at the same level of theory. The imaginary, reactive vibration is shown below. The bond forming process is synchronized upon observation, as would be expected for a concerted cycloaddition reaction.

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:45:48Z

Lh2213: /* Observing the Regioselectivity of the Diels-Alder Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
An initial C-C guess of 2.5000 angstroms was used in a Berny optimisation of the Diels-alder reaction. This time, the ENDO/EXO products were assesed using maleic annhydride and cyclohexa-1,3 diene.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 56:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 57:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 58:''' EXO-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 59: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 60: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 61: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:37:55Z

Lh2213: /* The EXO Product */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|-
|Partially formed C-C
|2.11925 / 2.11928
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths. The Van der Walls radius of a Carbon atom is 1.7 Å. Therefore, the partially formed bond is shorter than 2 times this distance, indicating strong interaction within the system.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|'''Fig 56:''' ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|'''Fig 57:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|'''Fig 58:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|'''Fig 59:'''ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|'''Fig 60:''' ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 61:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 62:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 63:''' EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|'''Fig 64:''' EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|'''Fig 65:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|'''Fig 66:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|'''Fig 67:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|'''Fig 68:''' EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 69: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 70: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}
'''Fig 71: '''Bond distances for the ENDO and EXO transition structures. r and r' denote the partly formed C-C bonds and the C-C through space distances between the -(CO)-O-(CO- fragment and either the C=C unit (ENDO) or the C-C unit (EXO).

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:21:38Z

Lh2213: /* The EXO Product */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|'''Fig 56:''' ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|'''Fig 57:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|'''Fig 58:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|'''Fig 59:'''ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|'''Fig 60:''' ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 61:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 62:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 63:''' EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|'''Fig 64:''' EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|'''Fig 65:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|'''Fig 66:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|'''Fig 67:''' EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|'''Fig 68:''' EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig 69: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 70: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:19:50Z

Lh2213: /* The EXO Transition State */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|'''Fig 56:''' ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|'''Fig 57:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|'''Fig 58:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|'''Fig 59:'''ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|'''Fig 60:''' ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|'''Fig 61:''' EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|'''Fig 62:''' EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|'''Fig 63:''' EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:18:51Z

Lh2213: /* The ENDO Product */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|'''Fig 56:''' ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|'''Fig 57:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|'''Fig 58:''' ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|'''Fig 59:'''ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|'''Fig 60:''' ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:17:13Z

Lh2213: /* The ENDO Transition State */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|'''Fig 51:''' ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|'''Fig 52:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|'''Fig 53:''' ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|'''Fig 54:''' ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|'''Fig 55:''' ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:14:20Z

Lh2213: /* Finding the Transition State Geometry of the Diels-Alder Prototype Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|'''Fig 42'''Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|'''Fig 43'''Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|'''Fig 44'''Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|'''Fig 45'''Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|'''Fig 46'''Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|'''Fig 47'''Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig 48: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig 49: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 50: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:11:24Z

Lh2213: /* Finding the Transition State Geometry of the Diels-Alder Prototype Reaction */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 42: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:07:42Z

Lh2213: /* The Diels-Alder Cycloaddition */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 42: '''Ethylene and cis-butadiene bond lengths.

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-26T00:02:40Z

Lh2213: /* Optimizing cis-Butadiene and Visualizing the HOMO and LUMO */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C (cis-Butadiene)
|1.39748
|}
'''Fig 42: '''Ethylene and cis-butadiene bond lengths.
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:53:38Z

Lh2213: /* Optimizing cis-Butadiene and Visualizing the HOMO and LUMO */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig 37:'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig 38:'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig 39:'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|'''Fig 40:''' Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|'''Fig 41:''' Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:51:19Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}
'''Fig 36: '''Activation energies calculated from Thermochemistry data.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:48:11Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 34'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 35'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:47:36Z

Lh2213: /* Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|'''Fig 35'''-234.55698303
File:OTCABTS(G)-BOAT.png|'''Fig 36'''-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:46:32Z

Lh2213: /* Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|'''Fig 30''' -231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|'''Fig 31'''-231.69166702
File:OTCABTS(F)II.png|'''Fig 32'''-231.69157910
File:OTCABTS(F)III.png|'''Fig 33'''-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:42:43Z

Lh2213: /* Optimizing to the Boat Transition Structure (QST2 Method) */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to a Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|-231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|-231.69166702
File:OTCABTS(F)II.png|-231.69157910
File:OTCABTS(F)III.png|-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:42:15Z

Lh2213: /* Optimizing to the Boat Transition Structure (QST2 Method) */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 25:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig 26''' Failed optimization to Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 27:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig 28''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig 29: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|-231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|-231.69166702
File:OTCABTS(F)II.png|-231.69157910
File:OTCABTS(F)III.png|-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:40:43Z

Lh2213: /* Vibrations and Thermochemistry Data */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K. These quantities were used in later calculations to determine the activation energies.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 12:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig''' Failed optimization to Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 12:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|-231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|-231.69166702
File:OTCABTS(F)II.png|-231.69157910
File:OTCABTS(F)III.png|-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:39:02Z

Lh2213: /* Optimizing to the Chair Transition Structure (Guess Structure Method) */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: <math>\omega=\sqrt{k\over m}</math> thus giving an imaginary output. The actual "-" is merely a notation, used by Gaussian, to define such frequencies.

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 12:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig''' Failed optimization to Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 12:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|-231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|-231.69166702
File:OTCABTS(F)II.png|-231.69157910
File:OTCABTS(F)III.png|-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}

Rep:Mod:L Harnett Trans Struct

2016-02-25T23:32:45Z

Lh2213: /* Optimizing to the Boat Transition Structure (QST2 Method) */

== Introduction ==
This study focuses on use of various calculation methods within Gausssian (Gaussview) to calculate the molecular geometries, ground state energies, vibrations and orbitals of the reactants, products, and transition states associated with the Cope Rearrangement and the Diels-Alder Cycloaddition.

Keywords: Antiperiplanar, Gauche, HOMO (Highest Occupied Molecular Orbital), LUMO (Lowest Occupied Molecular Orbital).

=== An Introduction to the Calculation Methods: HF, DFT and Semi-Empirical and their Basis Sets ===
Three main calculation types were used within this study: HF (Hartree-Fock), DFT (Density Functional Theory) and the Semi-Empirical method Austin Model 1 (AM1). These were used either comparatively, to observe how differences in the method result in different outputs (e.g. ground state energy values), or out of necessity with regards to the required level of detail of the calculation. All of them operate under the same principle in that the Time Independent Schrodinger Equation (TISE) must be solved for the system in question. Thus producing information regarding the wave function of the system from which the other properties of the system can be derived.

<math>\hat{H}\psi_i=E_i\psi_i</math>
* <math>\hat{H}</math> is the an Operator which corresponds to the specific method used in the calculation (HF, DFT, AM1...).
* <math>\psi_i</math> is the eigen-function used to describe the molecular wave-function which is derived from the basis set used in the calculation (STO-3G, 3-21G, 6-31G, 6-311G...).
* <math>E_i</math> is the output energy eigenvalue of the system.

==== The HF Calculation Method ====
The Hartree-Fock method of calculation uses a Slater Determinant to approximate the molecular wave function, formed from a basis set of N-orbitals. For these N-orbitals, N-coupled equations are derived, the solution of which produces the HF wave function and the energy of the system.

<math>\hat{F}(1)=\hat{H}^{core}(1)+\sum_{j=1}^{n/2}[2\hat{J}_j(1)-\hat{K}_j(1)]</math>
* <math>\hat{F}(1)</math> is the Fock Operator for one electron.
* <math>\hat{H}^{core}(1)</math> is the Hamiltonian Operator corresponding to a single, core electron.
* <math>\hat{J}_j(1)</math> is the Coulomb Operator (potential energy between an electron(1) and an electron of density <math>-|\psi_j(2)|^2</math>.
* <math>\hat{K}_j(1)</math> is the Exchange Operator which satisfies the condition that the wave-function must be anti-symmetric with respect to particle exchange.
<nowiki>http://www.sciencedirect.com/science/article/pii/0010465587900531</nowiki>

==== The DFT Calculation Method ====

<math>E_{DFT}[\rho]=T_0+\int v(r)rho(r)\, dr^3+J[\rho]+E_{xc}[\rho]</math>
* <math>E_{DFT}[\rho]</math> is the electronic energy as a function of the electron density.
* <math>\rho(r)</math> is the total electron density at a point <math>r</math>
* <math>T_0</math> is the kinetic energy of the electrons in the system.
* <math>\int v(r)rho(r)\, dr^3</math> represents the electron-nuclear attraction.
* <math>J[\rho]</math> represents the electron-electron attraction.
* <math>E_{xc}[\rho]</math> is the Exchange Correlation Potential due to the electron spin and charge.

==== The Semi-Empirical AM1 Calculation Method ====

==== The Basis Sets ====
For Both the Hartree-Fock

== The Cope Rearrangement ==

=== Optimizing the Reactants and Products ===
Before investigating the transition structure geometries and energies, the optimized reactants and products were assessed in order to validate the results of later calculations (SEE: Finding the Lowest Energy Conformer, via the Transition State, using the IRC Method).

==== Comparison of the Energies of the Anti and Gauche Conformers ====
Both the Antiperiplanar (anti2) and Gauche (gauche3) conformers were optimized under the HF/3-21G level of theory. <gallery widths=300px heights=200px>
File:OTRAP(A)-newman.png|'''Fig 1:''' Antiperiplanar Conformer of 1,5-Hexadiene, Total Energy: -231.69253528 a.u., Point Group: Ci
File:OTRAP(B)-newman.png|'''Fig 2:''' Gauche Conformer of 1,5-Hexadiene, Total Energy: -231.69266120 a.u., Point Group: C1
</gallery>From purely steric considerations, the antiperiplanar conformer was expected to be of lower energy, and therefore more stable, due to a predicted clash between the ethylene groups along the dihedral axis (SEE: Figures 1 and 2). However upon comparison of the energies, the Gauche conformer was observed to be lower in energy. Visualization of the HOMOs and LUMOs revealed a secondary π-interaction (between the π and π* orbitals respectively) exclusively within the Gauche conformer (SEE: Figures 3-8). Thus indicating a stabilizing electronic interaction that overrules any potential destabilization resulting from steric interactions. <gallery widths=300px heights=200px>
File:OTRAP(C)i.png|'''Fig 3:''' Gauche Conformer (side on representation)
File:OTRAP(C)iii.png|'''Fig 4:''' Gauche Conformer HOMO
File:OTRAP(C)ii.png|'''Fig 5:''' Gauche Conformer LUMO
</gallery>

<gallery widths=300px heights=200px>
File:OTRAP(C)iv.png|'''Fig 6:''' Antiperiplanar Conformer (side on representation)
File:OTRAP(C)v.png|'''Fig 7:''' Antiperiplanar Conformer HOMO
File:OTRAP(C)vi.png|'''Fig 8:''' Antiperiplanar Conformer LUMO
</gallery>In addition to this π-π interaction, previous studies have also provided evidence of a CH-π interaction: http://pubs.rsc.org/en/content/articlepdf/2004/ce/b313104a. However, such an interaction was not observed in this study.

==== Comparison of B3LYP/6-31G* to the HF/3-21G Level of Theory ====
The HF optimized antiperiplanar conformer was re-optimized under the B3LYP/6-31G* level of theory. Visually, this appeared to have no effect on the overall morphology of the molecule (Comparing Figures 1 and 9) . However, the energies as well as the molecular geometry (bond lengths and angles) were altered. The data regarding these findings is shown below (SEE: Figure 10). It is important to note that direct comparison of the energies across different calculation types (e.g. HF and DFT) is physically meaningless due to differences it in the theoretical constraints placed upon them.

<gallery widths=300px heights=200px>
File:OTRAP(F)-newman.png|'''Fig 9:''' Antiperiplanar Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61171063 Hartrees, Point Group: Ci
File:OTRAP(F)-NUM.png|'''Fig 10:''' Numbered atom diagram of Antiperiplanar 1,5-Hexadiene
File:Gauche 631g.png|'''Fig 11:''' Gauche Conformer of 1,5-Hexadiene (DFT Calculation), Total Energy: -234.61132934 Hartrees Point Group: C1
File:Gauche 631g-labelled.png|'''Fig 12:''' Numbered atom diagram of Gauche 1,5-Hexadiene
</gallery>

==== Antiperiplanar Conformer Geometry ====
The Ci symmetry of the Antiperiplanar conformer, resulted in many of its geometric quantities being equivalent.
{| class="wikitable"
!Bond
!DFT Length / Angstroms
!HF Length / Angstroms
|-
|11-7/9-14
|1.33824
|1.31613
|-
|7-4/1-9
|1.50720
|1.50891
|-
|4-1
|1.55504
|1.55275
|}
'''Fig 13: '''Antiperiplanar conformer bond lengths. More variation can be seen in the C=C bonds between the two levels of theory. Indicating that the HF produces a good approximation of the σ framework, but the π system is better approximated by a higher level DFT calculation.
{| class="wikitable"
!Angle
!DFT / θ°
!HF / θ°
|-
|13-11-12/15-14-16
|116.291
|116.310
|-
|11-7-8/14-9-10
|119.213
|119.680
|-
|8-7-4/10-9-1
|115.563
|115.507
|-
|11-7-4/14-9-1
|125.220
|124.806
|-
|6-4-5/3-1-2
|106.708
|107.715
|-
|7-4-1/9-1-4
|112.648
|111.349
|-
|7-4-1-9/5-4-1-3/6-4-1-2
|180.000
|180.000
|-
|8-7-4-5/3-1-9-10
|177.177
|174.269
|}
'''Fig 14: '''Antiperiplanar conformer bond angles. Some small changes in magnitude, but the overall distribution remained the same.

==== Gauche Conformer Geometry ====
More bond lengths and angles needed to be collected, compared to the Antiperiplanar conformer, due to the lack of symmetry in the Gauche conformer (Point group: C1)
{| class="wikitable"
!Bond
!DFT or HF
!Length
|-
|11-7
|HF
|1.31629
|-
|11-7
|DFT
|1.33382
|-
|7-1
|HF
|1.50847
|-
|7-1
|DFT
|1.50490
|-
|1-4
|HF
|1.55314
|-
|1-4
|DFT
|1.55013
|-
|4-9
|HF
|1.50929
|-
|4-9
|DFT
|1.50438
|-
|9-14
|HF
|1.31646
|-
|9-14
|DFT
|1.33360
|}
'''Fig 15: '''Gauche Conformer bond lengths. The same conclusions as in Figure 13 can be drawn from here also.
{| class="wikitable"
!Location
!DFT or HF
!θ°
|-
|11-7-1
|HF
|124.530
|-
|11-7-1
|DFT
|125.478
|-
|4-9-14
|HF
|125.024
|-
|4-9-14
|DFT
|124.934
|-
|7-1-4-9
|HF
|67.664
|-
|7-1-4-9
|DFT
|66.323
|}
'''Fig 16: '''Gauche conformer bond angles.

==== Vibrations and Thermochemistry Data ====
A frequency analysis of the antiperiplanar conformer was also carried out undef the B3LYP/6-31G* regime. No imaginary frequencies were found, indicating a stable structure.

<gallery widths=600px heights=200px>
File:OTRAP(G).png|'''Fig 17:''' Predicted IR Spectrum based on the analysis of the vibrations of the antiperiplanar conformer 1,5-Hexadiene
</gallery>

{| class="wikitable"
!Quantity
!Value
|-
|Sum of electronic and zero-point Energies
|'''-234.469219'''
|-
|Sum of electronic and thermal Energies
|'''-234.461869'''
|-
|Sum of electronic and thermal Enthalpies
|'''-234.460925'''
|-
|Sum of electronic and thermal Free Energies
|'''-234.500809'''
|}
'''Fig 18: '''Thermochemistry data for the antiperiplanar conformer of 1,5- hexadiene. The "sum of the electronic and zero point energies" term accurately described the energy of the system at 0 K, whilst the "sum of electronic and thermal energies term" accurately describes the energy of the system at 298.15 K.

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTRAP(A).LOG]]
|HF/3-21G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(B).LOG]]
|HF/3-21G optimised structure of gauche 1,5-Hexadiene
|-
|[[File:OTRAP(F).LOG]]
|B3LYP/6-31G optimised structure of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G).LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene
|-
|[[File:OTRAP(G)_0K.LOG]]
|B3LYP/6-31G frequency analysis of antiperiplanar 1,5-Hexadiene at 0 K
|-
|[[File:GAUCHE 631G.LOG]]
|B3LYP/6-31G frequency analysis of gauche 1,5-Hexadiene
|}

=== Optimizing the "Chair" and "Boat" Transition Structures ===
The Chair and Boat transition structures were optimized using three different methods:
* The Berny "Guess Structure" method
* The Frozen Coordinate method
* The QST2 method

==== Optimizing to the Chair Transition Structure (Guess Structure Method) ====
Two HF/3-21G optimized allyl fragments, positioned against one another in an approximate Chair-like conformation. With the end to end (where the Cope rearrangement bond breaks/forms) distances set to 2.2 Å. The structure was then optimized to the transition state (using the Berny calculation option). A frequency analysis was also carried out on the optimized structure to simulate the molecular vibrations. A single imaginary frequency was found. which corresponded directly to the molecular motion associated with the cope rearrangement. This vibration is defined as imaginary due to it's force constant value in the quantum harmonic oscillator. If the second derivative along a potential energy surface is negative at the maximum, the force constant derived will be negative. The angular frequency is obtained via: ω=

<gallery widths=300px heights=200px>
File:OTCABTS(A).png|'''Fig 19:''' Allyl fragment used to form half of the transition state.
File:OTCABTS(A)Chair.png|'''Fig 20:''' Allyl fragments arranged to form an approximate chair structure.
</gallery>

jmol>
<jmolApplet>
<title>Fig 21: Molecular vibration corresponding to the Cope Rearrangement (-817.62 cm^-1)</title><color>white</color>
<script>frame 19;vibration 1
</script><uploadedFileContents>OTCABTS(B).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Optimizing to the Chair Transition Structure (Frozen Coordinate Method) ====
If the guess structure were to fail, due to the initial "guess" being far from the correct geometry, the Frozen Coordinate method may be used to correct such an error. The fragments are "frozen" at the assumed end-end distances. Each fragment is then optimized (to a minimum) around this frozen geometry to produce a structure that is closer to the desired output. The end-end coordinates are then unfrozen and optimized the the transition structure, but from a corrected starting geometry. This was attempted on the same system used in the Guess Structure.

<gallery widths=300px heights=200px>
File:OTCABTS(C).png|'''Fig 22:''' Half-optimized Chair transition structure. The allyl fragments are optimized to a minimum whilst the end-end distance frozen at 2.2 Angstroms
File:OTCABTS(D).png|'''Fig 23:''' Fully optimized Chair transition structure with the now unfrozen bonds optimized independently of the individual allyl fragments.
</gallery>

Upon comparison to the Guess Structure results, the Frozen Coordinate method effectively produces the same energy and and end-end distances. The Frozen Coordinate end-end distances are much similar and relate more to the expected symmetry of the transition structure. This is most likely the result of the corrected geometry of the frozen coordinate method.
{| class="wikitable"
!Method
!Energy / Hartrees
!End-end distance 1 / Angstroms
!End-end distance 2 / Angstroms
|-
|Guess
|'''-231.61932209'''
|'''2.01994'''
|'''2.02081'''
|-
|Frozen Coordinate
|'''-231.61932231'''
|'''2.02071'''
|'''2.02064'''
|}
'''Fig 24: '''Comparison of the energies and end-end distances of the Guess Structure method and the Frozen Coordinate method.

==== Optimizing to the Boat Transition Structure (QST2 Method) ====
Although the previous calculation types may have been appropriate for the Boat structure, the QST2 method was used. This method works by specifying the reactants and products of the reaction, with each of the individual atoms labelled consistently. The transition structure is then determined by interpolating between the two structures. However, the reactant and product geometries must be carefully considered as this method does not consider the twisting of intramolecular bonds. This is demonstrated in the Boat transition structure calculations. When the simple, antiperiplanar conformer is used, the QST2 calculation did not rotate the molecule in accordance with the Cope Rearrangement. Rather, it translated one of the allyl fragments such that a Chair-like conformation is generated. In order to avoid this, the molecule was twisted manually before the calculation. This brought the molecule closer to the desired geometry. The vibrations were simulated, via a subsequent frequency analysis, giving a single imaginary frequency corresponding to the Cope Rearrangement.

<gallery widths=300px heights=200px>
File:OTCABTS(E).png|'''Fig 12:''' Antiperiplanar conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product.png|'''Fig''' Failed optimization to Chair-like transition structure
File:OTCABTS(E)-twist.png|'''Fig 12:''' Twisted conformer of 1,5-Hexadiene
File:OTCABTS(E)-Product2.png|'''Fig''' Successful Boat transition structure
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-839.94 cm^-1)</title><color>white</color>
<script>frame 41;vibration 1
</script><uploadedFileContents>OTCABTS(E)TWIST.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Determining the Lowest Energy Conformer, via the Transition State, using the IRC Method ====

<gallery widths=300px heights=200px>
File:OTCABTS(F).png|-231.69157910
</gallery>

<gallery widths=300px heights=200px>
File:OTCABTS(F)I.png|-231.69166702
File:OTCABTS(F)II.png|-231.69157910
File:OTCABTS(F)III.png|-231.68295296
</gallery>All C2 except f(iii)-but flawed

==== Re-optimizing the Chair and Boat Transition Structures via the B3LYP/6-31G* Method and Calculating the Activation Energies ====

<gallery widths=300px heights=200px>
File:OTCABTS(G)-CHAIR.png|-234.55698303
File:OTCABTS(G)-BOAT.png|-234.54309307
</gallery>
{| class="wikitable"
!Conformer
!Reagent Energy (0 K) / hartrees
!Reagent Energy (298.15 K) / hartrees
!Transition Structure Energy (0 K) / hartrees
!Transition Structure Energy (298.15 K) / hartrees
!Activation Energy (0 K) / hartrees
!Activation Energy (298.15 K) / hartrees
|-
|Chair HF/3-21G
|
|
|'''-231.61932209'''
|
|
|
|-
|Chair B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.55698303'''
|
|
|
|-
|Boat HF/3-21G
|
|
|'''-231.60280246'''
|
|
|
|-
|Boat B3LYP/6-31G*
|<nowiki>-234.469219</nowiki>
|<nowiki>-234.461869</nowiki>
|'''-234.54309307'''
|
|
|
|}

==== Associated Output Files ====
{| class="wikitable"
!File
!Description
|-
|[[File:OTCABTS(A).LOG]]
|HF/3-21G optimised allyl fragment
|-
|[[File:OTCABTS(B).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(C).LOG]]
|HF/3-21G half-optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(D).LOG]]
|HF/3-21G optimised chair transition structure for the Cope Rearrangement (Frozen Coordinate Method)
|-
|[[File:OTCABTS(E).LOG]]
|Failed HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(E)-BERNY.LOG]]
|HF/3-21G optimised boat transition structure for the Cope Rearrangement (Berny Method)
|-
|[[File:OTCABTS(E)TWIST.LOG]]
|Successful HF/3-21G optimised boat transition structure for the Cope Rearrangement (QST2 Method)
|-
|[[File:OTCABTS(F).LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)I.LOG]]
|HF/3-21G IRC output optimisation for the chair transition structure (Berny) of the Cope Rearrangement
|-
|[[File:OTCABTS(F)II.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (increased number of points)
|-
|[[File:OTCABTS(F)III.LOG]]
|HF/3-21G IRC calculation for the chair transition structure (Berny) of the Cope Rearrangement (computing force constants at every step)
|}

== The Diels-Alder Cycloaddition ==
The Semi-Empirical, AM1 calculation type was used for all of the folowing calculations.

==== Optimizing ''cis''-Butadiene and Visualizing the HOMO and LUMO ====
Before forming the transition states of the Diels-Alder reaction, the two reagents: ethylene and cis-butadiene were optimized and their HOMOs and LUMOs plotted. This aided the visualization and characterization steps that followed.<gallery widths=300px heights=200px>
File:TDAC(I).png|'''Fig'''Cis-Butadiene
File:TDAC(I)-HOMO.png|'''Fig'''HOMO of cis-Butadiene. Antisymmetric with respect to the plane.
File:TDAC(I)-LUMO.png|'''Fig'''LUMO of cis-Butadiene. Symmetric with respect to the plane
</gallery>

<gallery widths=300px heights=200px>
File:Ethylene-homo.png|Ethylene HOMO. Symmetric with respect to the plane.
File:Ethylene-lumo.png|Ethylene LUMO. Antisymmetric with respect to the plane.
</gallery>
{| class="wikitable"
!Bond
!Length
|-
|C=C (Ethylene)
|1.38297
|-
|C=C (cis-Butadiene)
|1.38195 / 1.38186
|-
|C-C
|1.39748
|}
{| class="wikitable"
!Bond
!Length (ENDO)
!Length (EXO)
|-
|<nowiki>-C=C- (Diene)</nowiki>
|1.39307 / 1.39306
|1.39442 / 1.39439
|-
|=C-C= (Diene)
|1.39725
|1.39679
|-
|<nowiki>-C-C- (Diene)</nowiki>
|1.52797
|1.52208
|-
|<nowiki>-C-C= (Diene)</nowiki>
|1.49053 / 1.49054
|1.48979 / 1.48970
|-
|<nowiki>-C=C- (Dieneophile)</nowiki>
|1.40850
|1.41012
|-
|=C-C= (Dieneophile)
|1.48923 / 1.48923
|1.48818 / 1.48817
|-
|<nowiki>-C=O (Dieneophile)</nowiki>
|1.22057 / 1.22057
|1.22054 / 1.22054
|-
|<nowiki>-C-O-(Dieneophile)</nowiki>
|1.40896 / 1.40896
|1.40963 / 1.40963
|-
|r
|2.16234 / 2.16242
|2.17050 / 2.17024
|-
|r'
|2.89221 / 2.89228
|2.94495 / 2.94496
|}

==== Finding the Transition State Geometry of the Diels-Alder Prototype Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

<gallery widths=300px heights=200px>
File:TDAC(II).png|Diels-Alder Cycloaddition transition structure
File:TDAC(II)-HOMO.png|Diels-Alder Cycloaddition transition structure(HOMO). The cis-butadiene HOMO interacts with the ethylene LUMO. (Antisymmetric)
File:TDAC(II)-LUMO.png|Diels-Alder Cycloaddition transition structure(LUMO). The cis-butadiene LUMO interacts with the ethylene HOMO. (Symmetric)
</gallery>

<gallery widths=300px heights=200px>
File:TDAC(II)-PRODUCT.png|Diels-Alder Cycloaddition Product
File:TDAC(II)-PRODUCT-HOMO.png|Diels-Alder Cycloaddition Product HOMO (Symmetric)
File:TDAC(II)-PRODUCT-LUMO.png|Diels-Alder Cycloaddition Product LUMO (Antisymmetric)
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Diels Alder Cycloaddition (-956.36 cm^-1, Synchronous)</title><color>white</color>
<script>frame 71;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: The lowest, positive molecular vibration corresponding to the Diels Alder Cycloaddition (147.38 cm^-1)</title><color>white</color>
<script>frame 72;vibration 1
</script><uploadedFileContents>TDAC(II).LOG</uploadedFileContents>
</jmolApplet>
</jmol>

==== Observing the Regioselectivity of the Diels-Alder Reaction ====
C-C guess: 2.5000 angstroms. (Bond length in optimized (AM1) product "" angstrom). Used Berny Method.

===== The ENDO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO.png|ENDO (-0.05150480 Hartrees)
File:TDAC(III)-ENDO-HOMO1.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO2.png|ENDO HOMO
File:TDAC(III)-ENDO-HOMO3.png|ENDO HOMO
File:TDAC(III)-ENDO-LUMO1.png|ENDO LUMO
</gallery>

===== The ENDO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-ENDO-PRODUCT.png|ENDO-PRODUCT
File:TDAC(III)-ENDO-PRODUCT-HOMO1.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO2.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-HOMO3.png|ENDO-PRODUCT-HOMO
File:TDAC(III)-ENDO-PRODUCT-LUMO.png|ENDO-PRODUCT-LUMO
</gallery>

===== The EXO Transition State =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO.png|EXO (-0.05041981 Hartrees)
File:TDAC(III)-EXO-HOMO.png|EXO-HOMO
File:TDAC(III)-EXO-LUMO.png|EXO-LUMO
</gallery>

===== The EXO Product =====
<gallery widths=300px heights=200px>
File:TDAC(III)-EXO-PRODUCT.png|EXO-PRODUCT
File:TDAC(III)-EXO-PRODUCT-HOMO1.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO2.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-HOMO3.png|EXO-PRODUCT-HOMO
File:TDAC(III)-EXO-PRODUCT-LUMO.png|EXO-PRODUCT-LUMO
</gallery>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-812.22 cm^-1)</title><color>white</color>
<script>frame 77;vibration 1
</script><uploadedFileContents>TDAC(III)-EXO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

jmol>
<jmolApplet>
<title>Fig: Molecular vibration corresponding to the Cope Rearrangement (-806.43 cm^-1)</title><color>white</color>
<script>frame 83;vibration 1
</script><uploadedFileContents>TDAC(III)-ENDO.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

===== Associated Output Files =====
{| class="wikitable"
!File
!Description
|-
|[[File:TDAC(I).LOG]]
|AM1 optimised cis-butadiene
|-
|[[File:TDAC(II).LOG]]
|AM1 optimised transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(II)-PRODUCT.LOG]]
|AM1 optimised product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-ENDO.LOG]]
|AM1 optimised endo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO.LOG]]
|AM1 optimised exo transition structure for the Diels-Alder Cycloaddition
|-
|[[File:TDAC(III)-EXO-PRODUCT.LOG]]
|AM1 optimised exo product of the Diels Alder Cycloaddition
|-
|[[File:TDAC(III)-PRODUCT.LOG]]
|AM1 optimised endo product of the Diels Alder Cycloaddition
|-
|[[File:ETHYLENE.LOG]]
|AM1 optimized ethylene molecule
|}