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Rep:Mod:als15TS

2018-02-14T09:15:30Z

Als15: /* Exercise 3: Xylylene and SO2 */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO, which is called variational theory. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in question as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math>,</center>

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate. B3LYP is a more expensive method than PM6, but yields greater accuracy.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2: Cyclohexadiene and 1,3-Dioxole==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product. The exo and endo reaction profiles overlap.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==References==

Rep:Mod:als15TS

2018-02-14T09:10:32Z

Als15: /* Exercise 2 */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO, which is called variational theory. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in question as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math>,</center>

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate. B3LYP is a more expensive method than PM6, but yields greater accuracy.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2: Cyclohexadiene and 1,3-Dioxole==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==References==

Rep:Mod:als15TS

2018-02-14T09:06:32Z

Als15: /* Appendix */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO, which is called variational theory. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in question as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math>,</center>

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate. B3LYP is a more expensive method than PM6, but yields greater accuracy.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==References==

Rep:Mod:als15TS

2018-02-14T08:52:32Z

Als15: /* PM6 and B3LYP [1] */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO, which is called variational theory. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in question as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math>,</center>

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate. B3LYP is a more expensive method than PM6, but yields greater accuracy.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-14T08:46:57Z

Als15: /* PM6 and B3LYP [1] */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO, which is called variational theory. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in question as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math>,</center>

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-14T08:41:15Z

Als15: /* PM6 and B3LYP [1] */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the Hartree-Fock model, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters, making it computationally more efficient but less accurate. B3LYP is a hybrid between the Hartree-Fock model and density functional theory, where an N-electron wavefunction can be expressed as one-electron densities by integrating the N-electron wave function over all spin and spatial coordinates apart from the spatial coordinates of the electron in questions as in:

<center><math>\rho(r_1) = N\int |\psi(x_1, x_2...x_N)|^2 ds_1 dx_2 ... dx_N</math></center>,

where ρ is the one-electron density, N is the number of electrons in the system, x is the spin and spatial coordinates of an electron and s is the spin coordinate.
</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-14T00:16:20Z

Als15: /* LOG Files */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

Exercise 3:
{| class="wikitable"
| Xylylene PM6
| [[Media:als15_XYLYLENE_MIN_PM6.LOG]]
|-
| SO<sub>2</sub> PM6
| [[Media:als15_SO2_MIN_PM6.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_EXO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXO_PRODUCT_MIN_PM6.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXO_TS_PM6_IRC.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_DAENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media: als15_DAENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Product PM6
| [[Media:als15_DAENDO_PRODUCT_MIN_PM6.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_DAENDO_TS_PM6_IRC.LOG]]
|-
| Chelotropic Guess TS PM6
| [[Media:als15_CHELO_FROZENTS_MIN_PM6.LOG]]
|-
| Chelotropic Transition State PM6
| [[Media:als15_CHELO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Chelotropic Product PM6
| [[Media:als15_CHELO_PRODUCT_MIN_PM6.LOG]]
|-
| Chelotropic IRC PM6
| [[Media: als15_CHELO_TS_IRC_PM6.LOG]]
|-
| Alternative 1 Guess TS PM6
| [[Media:als15_ALTLEFT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 1 Transition State PM6
| [[Media:als15_ALTLEFT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 1 Product PM6
| [[Media: als15_ALTLEFT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 1 IRC PM6
| [[Media:als15_ALTLEFT_TS_PM6_IRC.LOG]]
|-
| Alternative 2 Guess TS PM6
| [[Media:als15_ALTRIGHT_FROZENTS_MIN_PM6.LOG]]
|-
| Alternative 2 Transition State PM6
| [[Media:als15_ALTRIGHT_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Alternative 2 Product PM6
| [[Media: als15_ALTRIGHT_PRODUCT_MIN_PM6.LOG]]
|-
| Alternative 2 IRC PM6
| [[Media:als15_ALTRIGHT_TS_PM6_IRC.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:52:23Z

Als15: /* LOG Files */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [[Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|-
| Exo Guess TS PM6
| [[Media:als15_EXO_FROZENTS_MIN_PM6.LOG]]
|-
| Exo Transition State PM6
| [[Media:als15_Exo_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Exo Transition State B3LYP
| [[Media:als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Exo IRC PM6
| [[Media:als15_EXOTS_PM6_IRC.LOG]]
|-
| Exo Product PM6
| [[Media:als15_EXOPRODUCT_MIN_PM6.LOG]]
|-
| Exo Product B3LYP
| [[Media:als15_EXOPRODUCT_MIN_B3LYP.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:48:05Z

Als15: /* LOG Files */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

Exercise 2:
{| class="wikitable"
| 1,3-Dioxole PM6
| [[Media:als15_13DIOXOLE_MIN_PM6.LOG]]
|-
| 1,3-Dioxole B3LYP
| [Media:als15_13DIOXOLE_MIN_B3LYP.LOG]]
|-
| Cyclohexadiene PM6
| [[Media:als15_CYCLOHEXADIENE_MIN_PM6.LOG]]
|-
| Cyclohexadiene B3LYP
| [[Media:als15_CYCLOHEXADIENE_MIN_B3LYP.LOG]]
|-
| Endo Guess TS PM6
| [[Media:als15_ENDO_FROZENTS_MIN_PM6.LOG]]
|-
| Endo Transition State PM6
| [[Media:als15_ENDO_GUESSTS_TSBERNY_PM6.LOG]]
|-
| Endo Transition State B3LYP
| [[Media:als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG]]
|-
| Endo IRC PM6
| [[Media:als15_ENDOTS_PM6_IRC.LOG]]
|-
| Endo Product PM6
| [[Media:als15_ENDOPRODUCT_MIN_PM6.LOG]]
|-
| Endo Product B3LYP
| [[Media:als15_ENDOPRODUCT_MIN_B3LYP.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:34:49Z

Als15: /* LOG Files */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6.LOG]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:33:45Z

Als15: /* LOG Files */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==

Exercise 1:
{| class="wikitable"
| Ethene PM6
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
|-
| Butadiene PM6
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
|-
| Guess TS PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6]]
|-
| Transition State PM6
| [[Media:als15_GUESSTS_TSBERNY_PM6]]
|-
| IRC PM6
| [[Media:als15_IRC_PM6.LOG]]
|-
| Cyclohexene PM6
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:29:54Z

Als15: /* References */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==LOG Files==
{| class="wikitable"
! Ethene PM6
! Butadiene PM6
! Guess TS PM6
! Transition State PM6
! IRC PM6
! Cyclohexene PM6
|-
| [[Media: als15_ALKENE_MIN_PM6.LOG]]
| [[Media: als15_BUTADIENE_MIN_PM6.LOG]]
| [[Media:als15_FROZENGUESSTS_MIN_PM6.LOG]]
| [[Media:als15_GUESSTS_TSBERNY_PM6]]
| [[Media:als15_IRC_PM6.LOG]]
| [[Media:als15_PRODUCT_MIN_PM6.LOG]]
|}

==Appendix==

Rep:Mod:als15TS

2018-02-13T23:18:28Z

Als15: /* Exercise 3: Xylylene and SO2 */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

==References==
==Appendix==

Rep:Mod:als15TS

2018-02-13T23:15:40Z

Als15: /* C-C Bond Lengths */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. p.159. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T23:13:57Z

Als15: /* Exercise 3: Xylylene and SO2 */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
<p>The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.</p>
<br clear=all>

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T23:06:31Z

Als15: /* Exercise 2 */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
<p>Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.</p>
<br clear=all>

Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level.
{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>

Table 7: Energies of MOs relative to others in the same approach (endo or exo).
{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
<p>Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.</p>
<br clear=all>

Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
<br clear=all>

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]
<br clear=all>

Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO.
{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:59:52Z

Als15: /* Exercise 1: Reaction of Butadiene with Ethylene */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only π electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode in Figure 3.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state showing bond formation is synchronous.</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:53:05Z

Als15: /* Introduction */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP [1]===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:49:02Z

Als15: /* Introduction */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP<ref>McDouall, J. (2013). Computational Quantum Chemistry: Molecular Structure and Properties in Silico. p. 1-16. Royal Society of Chemistry.</ref>===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:37:37Z

Als15: /* Introduction */

==Introduction==
===Potential Energy Surfaces===
<p>A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.</p>

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:31:22Z

Als15: /* PM6 and B3LYP */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.
In the semi-empirical method, PM6, some integrals are omitted and replaced with empirical parameters.</p>

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:11:07Z

Als15: /* PM6 and B3LYP */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^{-ar} </math> or gaussian-type orbitals of the form <math> e^{-ar^2} </math> are used as they are easier to integrate, with gaussian-type orbitals being the easiest.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T22:09:29Z

Als15: /* PM6 and B3LYP */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\phi_{i}</math>, where the coefficients are found by minimising the expectation value of energy of the MO. In practice, instead of using AOs from the solution of the Schroedinger equation for the hydrogen atom, slater-type orbitals of the form <math> e^-ar </math> or gaussian-type orbitals of the form <math> e^-ar^2 </math> are used as they are easier to integrate, with gaussian-type orbital being the easiest.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T21:46:56Z

Als15: /* PM6 and B3LYP */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi =</math> <math>\sum c_{i}</math> <math>\fi_{i}</math>.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T21:45:49Z

Als15: /* Introduction */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

===PM6 and B3LYP===
<p> In the variational method, an MO is expressed as a linear combination of AOs, <math>\psi = \sum c_{i} \fi_{i}</math>.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T11:23:01Z

Als15: /* Exercise 3: Xylylene and SO2 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alternative 1 and alternative 2 reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T11:21:24Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3: Xylylene and SO<sub>2</sub>==
The possible reactions between xylylene and SO<sub>2</sub> are shown in Figure 6
and the reaction barriers and reaction energies are shown in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy, which makes sense as the bonding and sterics in the transition states and the products are similar. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions, the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 7-9 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure 6) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure 6: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 7: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 8: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 9: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alternative 1
! Alternative 2
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 10: Reaction profile of the exo, endo and chelotropic reaction, going from reactant to product.]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T11:02:27Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product. The two reactions of SO<sub>2</sub> with the alternative diene (Alternative 1 and 2 in Figure) are the most kinetically unfavourable and the most thermodynamically unfavourable due to the steric hindrance in the transition states and the fact that the six-membered ring is not aromatised.

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T10:44:53Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
[[File:Als15_reaction_scheme_exercise3.png|thumb|600px|left|Figure: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product.

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T10:44:25Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
[[File:Als15_reaction_scheme_exercise3.png|thumb|400px|left|Figure: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
<br clear=all>

The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product.

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-13T10:43:43Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
[[File:Als15_reaction_scheme_exercise3.png|thumb|left|Figure: Reaction schemes of possible reactions between xylylene and SO<sub>2</sub>]]
The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product.

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

File:Als15 reaction scheme exercise3.png

2018-02-13T10:40:59Z

Als15:

Rep:Mod:als15TS

2018-02-13T08:45:45Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. The more stable product from the chelotropic reaction is due to the S=O bond being stronger than a C-S bond and S-O bond combined. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product.

https://notendur.hi.is/agust/rannsoknir/papers/2010-91-CRC-BDEs-Tables.pdf
http://www.wiredchemist.com/chemistry/data/bond_energies_lengths.html
[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-12T15:47:48Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
The reaction barriers and reaction energies have been calculated for multiple possible reactions between xylylene and SO<sub>2</sub>, as seen in Table 10. The exo and endo reactions have the same reaction barrier and reaction energy. The chelotropic reaction is thermodynamically more favourable, but kinetically less favourable, than the exo and endo reactions. In all three of these reactions the reaction energies are negative, which is due to the aromatisation of the six-membered ring in xylylene, as shown in the IRCs in Figure 6-8 by the bond order of 1.5 in the product.

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

Table 10: Energies of the reactants, transition states and products of the exo, endo, chelotropic, alt-left and alt-right reaction between xylylene and SO<sub>2</sub>.

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-12T14:55:27Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|500px|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-12T14:54:10Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_reaction_profile.png|thumb|left|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

<br clear=all>

Rep:Mod:als15TS

2018-02-12T14:53:00Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}
<br clear=all>

[[File:als15_reaction_profile.png|thumb|left|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

Rep:Mod:als15TS

2018-02-12T14:51:03Z

Als15: /* Exercise 3 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

[[File:als15_secondary orbital interactions.png|thumb|left|Figure 9: Reaction profile of the exo, endo and chelotropic reaction]]

File:Als15 reaction profile.png

2018-02-12T14:43:29Z

Als15:

Rep:Mod:als15TS

2018-02-12T14:25:24Z

Als15: /* IRCs */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
<p>Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
<p>Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub></p>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
<p>Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub></p>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T14:24:39Z

Als15: /* IRCs */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===

[[File:als15_exo_IRC3.gif]]
Figure 6: IRC of the exo reaction between xylylene and SO<sub>2</sub>

[[File:als15_daendo_ts_pm6_IRC_2.gif]]
Figure 7: IRC of the endo reaction between xylylene and SO<sub>2</sub>

[[File:als15_chelo_ts_IRC_pm6_2.gif]]
Figure 8: IRC of the chelotropic reaction between xylylene and SO<sub>2</sub>

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T14:22:40Z

Als15: /* IRCs */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif|thumb|left]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif|thumb|left]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif|thumb|left]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T14:00:57Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table 9 and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T14:00:09Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

[[File: Als15_secondary_orbital_interactions.png|thumb|left|Figure 5: Primary and secondary orbital interactions in the HOMO of the endo transiton state.]]

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

File:Als15 secondary orbital interactions.png

2018-02-12T13:55:09Z

Als15:

Rep:Mod:als15TS

2018-02-12T11:34:26Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}
Table 9: HOMOs of the exo and endo transition states. There are secondary orbital interactions in the endo TS HOMO

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T11:31:32Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

{| class="wikitable"
! EXO HOMO-1 (MO40)
! EXO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|-
! ENDO HOMO-1 (MO40)
! ENDO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

{| class="wikitable"
! HOMO of Exo TS
! HOMO of Endo TS
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T11:25:06Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table and Figure 5. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

{| class="wikitable"
! EXO HOMO-1 (MO40)
! EXO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|-
! ENDO HOMO-1 (MO40)
! ENDO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-12T11:18:43Z

Als15: /* Thermochemistry */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction. The endo reaction is kinetically more favourable due to secondary orbital interactions. In the HOMO of the endo transition state the oxygen p-orbitals interact in-phase with carbon p-orbitals in the cyclohexadiene, as can be seen in Table and Figure. These secondary orbital interactions lower the energy of the HOMO of the endo transition state. So, in comparison to the exo transtion state - where secondary orbital interactions are absent, the energy of the endo transition state and ,therefore kinetic barrier, is lower.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

{| class="wikitable"
! EXO HOMO-1 (MO40)
! EXO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|-
! ENDO HOMO-1 (MO40)
! ENDO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}

Rep:Mod:als15TS

2018-02-11T20:38:16Z

Als15: /* Exercise 2 */

==Introduction==
A potential energy surface, PES, is a plot of potential energy, V, as a function of nuclear separation, r. In the case of a non-linear molecule, there are 3N-6 degrees of freedom, where N is the number of atoms, and there are 3N-5 degrees of freedom in a linear molecule. Minima, where configurations of reactants and products lie, are defined mathematically as a stationary point, <math>\frac{\partial V}{\partial r_i}=0</math>, where all the second derivatives are positive, <math>\frac{\partial^2 V}{\partial r_i^2}>0</math>. A transition state is a maximum in a minimum energy path between two minima. Mathematically, they are defined as stationary points where only one of the second derivatives is negative; the rest being positive.

===Vibrational Frequencies===
<p>The vibrational wavenumber, <math>\bar{\nu}</math>, is related to the force constant, k, in the equation:</p>
<center><math>\bar{\nu}=</math><math>\frac{1}{2 \pi c}</math><math>\sqrt{\frac{k}{\mu}}</math>,</center>
where c is the speed of light in vacuum, μ is the reduced mass of the molecule and k is the force constant, which is equal to the second derivative in a PES.
A structure at a minimum would therefore have all positive vibrational frequencies and a transition state structure would have one negative vibrational frequency. If there are more than one negative vibrational frequencies, the structure has a maximum potential energy in more than one dimension and this means there is a lower energy path connecting the two minima (via the transition state).
When the transition state is found, a reaction profile - energy versus the degree of freedom where the TS is at a maximum - can be plotted.

==Exercise 1: Reaction of Butadiene with Ethylene==
===MO Interaction===
<p>Figure 1 shows the interactions between the MOs of butadiene and ethene and the resulting MOs of the transition state. The MOs and relative energies of the reactants are a result of Huckel theory, where only pi electrons are considered. The computed MOs of the transition state, MO16-MO19, show that interactions between HOMO(diene)-LUMO(dienophile) and HOMO(dienophile)-LUMO(diene) both occur. This happens due to the similar energies and same symmetry of the MOs in each pair. If one MO was anti-symmetric and the other symmetric, the orbital overlap would be zero as there would be equal amounts of bonding interaction as there would be anti-bonding interaction. Any overlap is cancelled and it leads to two non-bonding orbitals, i.e. no interaction between orbitals of opposite symmetry. A pair of MOs that are both symmetric or both asymmetric results in a non-zero overlap, and hence interaction occurs. </p>

[[File:als15_MOdiagram2.png|left|thumb|600px|Figure 1: MO diagram of the reaction between butadiene and ethene]]
<br clear=all>
Table 1: HOMOs and LUMOs of butadiene and ethene
{| class="wikitable"
!
! Butadiene
! Ethene
|-
! HOMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 11; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 6; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! LUMO
| <jmol><jmolApplet>
<uploadedFileContents>Als15_BUTADIENE_MIN_PM6.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 58; mo 12; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ALKENE_MIN_PM6.LOG</uploadedFileContents>
<script>frame 12; mo 7; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

Table 2: MOs in transition state of the reaction between butadiene and ethene
{| class="wikitable"
! MO16 (HOMO-1)
! MO17 (HOMO)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 16; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 17; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|-
! MO18 (LUMO)
! MO19 (LUMO+1)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 18; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script>frame 16; mo 19; mo nodots nomesh fill translucent; mo titleformat"";mo cutoff 0.02</script>
<size>250</size>
<color>white</color>
</jmolApplet></jmol>
|}

===C-C Bond Lengths===
Table 3 shows how the carbon-carbon bond lengths, referred to in Figure 2, change as the reaction progresses. The bond lengths in the reactants and product are as expected when compared to typical carbon-carbon bond lengths of particular hybridisations, as seen in Table 4. The bond lengths in the transition state are more similar to that in the reactant, which suggests an early transition state. Furthermore, in the transition state, C1-C6 and C4-C5 are shorter than double the van der Waals radius of carbon (3.40Å), which means the two bonds have partially formed in the transition state. They are also the same length, which suggests bond formation is synchronous, as can be seen in the negative frequency vibrational mode below.

[[File:als15_bondlengths.png| 400px|left|thumb|Figure 2: Labelled carbons]]

<br clear=all>
Table 3: Bond lengths as the reaction progresses
{| class="wikitable"
!
! colspan="3" |Bond Length / Å
|-
!
! Reactants
! Transition State
! Product
|-
! C1-C2
| 1.33349
| 1.37979
| 1.50087
|-
! C2-C3
| 1.47077
| 1.41109
| 1.33698
|-
! C3-C4
| 1.33343
| 1.37977
| 1.50079
|-
! C4-C5
| -
| 2.11482
| 1.53717
|-
! C5-C6
| 1.32741
| 1.38177
| 1.53467
|-
! C6-C1
| -
| 2.11466
| 1.53708
|}
<br clear=all>

Table 4: Typical C-C bond lengths <ref>Fox, Marye Anne; Whitesell, James K. (1995). Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen. Springer. ISBN 978-3-86025-249-9.</ref>
{| class="wikitable"
!
! Typical Length / Å
|-
! C(sp3)-C(sp3)
| 1.54
|-
! C(sp2)-C(sp3)
| 1.50
|-
! C(sp2)-C(sp2)
| 1.47
|-
! C(sp2)=C(sp2)
| 1.34
|-
! C van der Waals radius
| 1.70
|}

<br clear=all>

<jmol><jmolApplet>
<uploadedFileContents>Als15_GUESSTS_TSBERNY_PM6.LOG</uploadedFileContents>
<script> frame 17; vibration 2 </script>
</jmolApplet></jmol>
<br clear=all>
<font size="2">Figure 3: Negative frequency of vibration at the transition state</font>

==Exercise 2==
===MO Diagram===
Table 5 and 6 show transition state MOs, optimised at the B3LYP level, for both the exo and endo approach respectively. For a given approach, the relative energies of the reactant MOs, as seen in Table 7, were found by running a single point energy calculation of the first frame of the IRC (optimised at a PM6 level). For both the exo and endo approach, the LUMO of the 1,3-dioxole is higher in energy than that of cyclohexadiene, and the same can be seen for the HOMOs. This is a sign of an inverse electron demand Diels-Alder reaction. The electron donating ether groups on the 1,3-dioxole increase the energy of the frontier MOs of 1,3-dioxole as it donates more electron density to the alkene via pi interactions between the O p-orbital and C p-orbitals in the alkene.

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 5: HOMO, HOMO-1, LUMO and LUMO-1 of exo transition state, optimised to B3LYP level

{| class="wikitable"
! HOMO-1 (MO40)
! HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|-
! LUMO (MO42)
! LUMO+1 (MO43)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 42; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 43; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.02</script>
</jmolApplet></jmol>
|}
<br clear=all>
Table 6: HOMO, HOMO-1, LUMO and LUMO-1 of endo transition state, optimised to B3LYP level

{| class="wikitable"
!
! Energy of MO (exo)/ a.u.
! Energy of MO (endo)/ a.u.
|-
! LUMO of 1,3-dioxole
| 0.02979
| 0.03219
|-
! LUMO of Cyclohexadiene
| 0.02111
| 0.02288
|-
! HOMO of 1,3-dioxole
| -0.32207
| -0.31696
|-
! HOMO of Cyclohexadiene
| -0.32217
| -0.32135
|}
<br clear=all>
Table 7: Energies of MOs relative to others in the same approach (endo or exo).

[[File:als15_exercise2_MOdiagram2.png|700px|left| thumb|Figure 4: MO diagram of the reaction between cyclohexadiene and 1,3-dioxole]]

<br clear=all>

===Thermochemistry===
Table 8 shows the reaction energies and reaction barriers for the endo and exo reaction between cyclohexadiene and 1,3-dioxole. The endo reaction is both kinetically and thermodynamically favourable, as it has a lower energy barrier and a more negative Gibbs free energy of reaction.
{| class="wikitable"
!
! Gibbs Free Energy / Hartrees
! Gibbs Free Energy / kJ mol<sup>-1</sup>
|-
! Cyclohexadiene
| -233.324375
| -612593.19323
|-
! 1,3-Dioxole
| -267.068637
| -701188.75986
|-
! Exo TS
| -500.329163
| -1313614.3175
|-
! Endo TS
| -500.332150
| -1313622.1599
|-
! Exo Product
| -500.417322
| -1313845.779
|-
! Endo Product
| -500.418692
| -1313849.3759
|-
! ΔG(exo)
| style="background: #c9c9c9"|
| -63.8259100
|-
! ΔG‡(exo)
| style="background: #c9c9c9"|
| 167.635590
|-
! ΔG(endo)
| style="background: #c9c9c9"|
| -67.4228100
|-
! ΔG‡(endo)
| style="background: #c9c9c9"|
| 159.793190
|}
Table 8: Gibbs free energy values at room temperature of structures optimised at B3LYP level

{| class="wikitable"
! EXO HOMO-1 (MO40)
! EXO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_EXO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|-
! ENDO HOMO-1 (MO40)
! ENDO HOMO (MO41)
|-
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 40; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
| <jmol><jmolApplet>
<uploadedFileContents>Als15_ENDO_GUESSTS_TSBERNY_B3LYP.LOG</uploadedFileContents>
<size>250</size>
<color>white</color>
<script>frame 50; mo 41; mo nodots nomesh fill translucent;mo titleformat"";mo cutoff 0.01</script>
</jmolApplet></jmol>
|}

==Exercise 3==
===IRCs===
<p>Exo</p>
[[File:als15_exo_IRC3.gif]]

<p>Endo</p>
[[File:als15_daendo_ts_pm6_IRC_2.gif]]

<p>Chelotropic</p>
[[File:als15_chelo_ts_IRC_pm6_2.gif]]

===Thermochemistry===
Energy in kj/mol, PM6:

{| class="wikitable"
!
! Exo
! Endo
! Chelotropic
! Alt-Left
! Alt-Right
|-
! Xylylene
| colspan="6" |467.266
|-
! SO2
| colspan="6" |-311.421
|-
! TS
| 241.746
| 241.746
| 260.087
| 267.985
| 275.822
|-
! Product
| 56.3275
| 56.3327
| 0.013128
| 172.256
| 176.709
|-
! ΔG
| -99.5175
| -99.5123
| -155.832
| 16.4110
| 20.8640
|-
! ΔG‡
| 85.9010
| 85.9010
| 104.242
| 112.140
| 119.977
|}