View source for Rep:Mod:ts cyy113 edit

{| class="wikitable"
! rowspan="2" |Species or IRC
!Basis Set
|-
!PM6
|-
|o-xylylene
|[[File:Xylene cyy113.log]]
|-
|SO<sub>2</sub>
|[[File:SO2 CYY113.LOG]]
|-
|Endo TS
|[[File:EX3ENDOTS CYY113.LOG]]
|-
|Exo TS
|[[File:XYLYLENE exoex3 TS cyy113.LOG]]
|-
|Cheletropic TS
|[[File:INDENE_TS_cyy113.LOG]]
|-
|Endo Product
|[[File:ENDOPRODUCT cyy113ex3.LOG]]
|-
|Exo Product
|[[File:EXOPRODUCT OPT CYY113.LOG]]
|-
|Cheletropic Product
|[[File:INDENEPRODUCT CYY113.LOG]]
|-
|Endo IRC
|[[File:EX3ENDOTSIRC cyy113.LOG]]
|-
|Exo IRC
|[[File:Xylene exo ts irccyy113.log]]
|-
|Cheletropic IRC
|[[File:INDENEIRC CYY113.LOG]]
|}
{| class="wikitable"
|+'''Table X: Energies of the reactants, TS and products in the alternative reaction between o-xylylene and SO<sub>2</sub>'''
! colspan="2" |Xylylene
! colspan="2" |SO<sub>2</sub>
! colspan="2" |Endo TS
|-
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.102070
|267.98480541
|-
! colspan="2" |Endo Product
! colspan="2" |Exo TS
! colspan="2" |Exo Product
|-
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
|-
|0.065609
|172.25644262
|0.105054
|275.81929801
|0.067306
|176.711916
|}
{| class="wikitable"
!Reaction
!Formation of Endo Product
!Formation of Exo Product
|-
|Reaction barrier/ kJ mol<sup>-1</sup>
|<nowiki>+110.1</nowiki>
|<nowiki>+117.9</nowiki>
|-
|Reaction energy/ kJ mol<sup>-1</sup>
|<nowiki>+14.4</nowiki>
|<nowiki>+18.8</nowiki>
|}
'''The relevant C-C bond lengths are tabulated below:'''
{| class="wikitable"
|+Table 2
!Reactant C-C bond lengths
!TS C-C bond lengths
!Product C-C bond lengths
|-
|[[File:Reactantbondlengths cyy113.PNG|frameless]]
|[[File:Tsbondlength cyy113.PNG|frameless]]
|[[File:Prodbondlength cyy113.PNG|frameless]]
|-
!Reported sp<sup>2</sup> and sp<sup>3 </sup>bond lengths
in cyclohexene<ref>J. F. Chiang and S. H. Bauer, ''Journal of the American Chemical Society'', 1969, '''91''', 1898–1901</ref> /Å
!Typical sp<sup>2</sup> and

sp<sup>3</sup> bond lengths<ref>E. V. Anslyn and D. A. Dougherty, ''Modern physical organic chemistry'', Univ. Science Books, Sausalito, CA, 2008, pp.22</ref> /Å
!Van Der Waals' radius 
of C atom<ref>G. S. Manku, ''Theoretical principles of inorganic chemistry'', Inter-India Publ., New Delhi, 1986, pp.97</ref> /Å
|-
|[[File:Literaturebondlengths cyy113.PNG|frameless]]
|sp<sup>2</sup> = 1.31-1.34

sp<sup>3</sup> = 1.53-1.55
|0.77
|}
<span style="color:#0000FF">Write whether the orbital overlap integral is zero or non-zero for the case of a symmetric-antisymmetric interaction, a symmetric-symmetric interaction and an antisymmetric-antisymmetric interaction.</span>

The orbital overlap integral <math>S</math> quantifies the extent of overlap between reactant MOs (denoted by <math>\psi_1</math> and <math>\psi^*_2</math>. It is given by the following equation.<ref>P. Atkins and J. De Paula, ''Atkins' Physical Chemistry'', University Press, Oxford, 10th edn., 2014.</ref>

<center><math>S=\int {\psi_1\psi^*_2 d\tau}</math></center>
<center><small> Equation 2: Equation of the overlap integral </small> </center>

The possible values of <math>S</math> depends on <math>\psi_1</math> and <math>\psi^*_2</math> and they are as follows:
{| class="wikitable"
!Type of interaction
|Symmetric-Antisymmetric
|Symmetric-Symmetric
|Antisymmetric-Antisymmetric
|-
!Value of S
|<center>Zero</center>
|<center>Non-Zero</center>
|<center>Non-Zero</center>
|}

== intro ==
<span style="color:#0000FF">In your introduction, briefly describe what is meant by a minimum and transition state in the context of a potential energy surface. What is the gradient and the curvature at each of these points? (for thought later on, how would a frequency calculation confirm a structure is at either of these points?) </span>

The dynamics of a chemical reaction can be investigated using a potential energy surface (PES) obtained by plotting <math>V(q_1,q_2)</math> against <math>q_1</math> and <math>q_2</math>, where <math>V(q_1,q_2)</math> is the potential energy and <math>q_1</math>, <math>q_2</math> are order parameters of a reaction. Reactants and products occur at the minimum of the PES while transition states occur at the saddle point. Mathematically, they are both stationary points, so the gradient (i.e. the first derivative <math>\partial V/\partial q_i</math>) of both the minimum and the saddle point is 0. However, the curvature of the minimum is concave while that of the saddle point is convex. Thus, they can only be distinguished by taking the second derivative <math>\partial^2 V/\partial q_i^2</math>, which will be positive for the minimum, but negative for the saddle point.

Since a chemical bond can be modelled as a spring, Hooke's Law (eq. 1) can be invoked, where <math>V</math> is the elastic potential energy, <math>k</math> is a constant and <math>x</math> is the displacement of the particle from its equilibrium position.

<center><math> V = 1/2 kx^2 </math></center>
<center><small> ''Equation 1: Hooke's Law'' </small></center>

Taking the second derivative of this equation results in eq. 2. Thus, <math>k</math> will be positive for all minimum points but negative for saddle points. Physically, this means that all coordinates have been minimised for reactants and products, but some coordinates have not been minimised for transition states. Thus, reactants and products remain stable unless they are excited with energy but transition states are inherently unstable and will rearrange into a stable form spontaneously.

<center><math> \partial^2 V/\partial x^2 = k </math></center>
<center><small> ''Equation 2: Second derivative of Hooke's Law'' </small></center>

Since the vibrational frequency <math>\omega</math> is a function of <math>\sqrt k</math> (eq. 3), reactants and products will always give real frequencies, while transition states will give imaginary frequencies. A well-chosen reaction coordinate will only result in a transition state with one imaginary frequency, as all other coordinates have been minimised.

<center><math> \omega = {1/2\pi}\sqrt {k/m} </math></center>
<center><small> ''Equation 3: Relation between frequency and <math>k</math>'' </small></center>

In these exercises, computational methods were used to determine suitable reaction coordinates of several [4+2] Diels-Alder cycloadditions. The coordinates of the reactants, transition state (TS) and products were optimised using Gaussian, from which an Intrinsic Reaction Coordinate (IRC) calculation was run to visualise the approach trajectory of the reactants to form a transition state and subsequently the product. Further observable parameters (Bond Length and Energies) were also analysed from the optimised coordinates and IRC calculation. The wavefunctions of the optimised species were visualised as molecular orbitals (MOs) and their shapes and symmetries were compared to theoretical predictions. 

Optimisation was conducted at the semi-empirical PM6 level first, before some calculations were refined using Density Functional Theory (DFT) methods at the BY3LP/6-31(d) level. The PM6 optimisation uses a fitted method drawing on experimental data, hence it is less accurate. However, it saves computational resources and is a good starting point for initial calculations. The BY3LYP/6-31(d) optimisation is more accurate but it comes at the cost of computational effort.


{|
|-
| <center>'''Formation of endo product''' </center> || <center>'''Formation of exo product'''</center>
|-
| <center>[[File:Endo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(A): Energy profile of the endo product formation]]</center> || <center>[[File:Exo_profilediagram_cyy113.PNG|centre|thumb|300x300px|(B): Energy profile of the exo product formation]]</center>
|-
| colspan="2" |<small>''Fig. Y: Energy profile diagrams showing the reaction barrier and reaction energies of the reaction''</small>
|}

== MO Analysis ==
<span style="color:#0000FF">Using your MO diagram for the Diels-Alder reaction, locate the occupied and unoccupied orbitals associated with the DA reaction for both TSs by symmetry. Find the relevant MOs and add them to your wiki (at an appropriate angle to show symmetry). Construct a new MO diagram using these new orbitals, adjusting energy levels as necessary. </span>

The MO diagrams for the formation of the endo and exo transition states, as well as the relevant MOs, are shown in Fig. 9 and Table 7 respectively. The relative ordering of the MO energies were based on the orbital energies obtained from a BY3LYP/6-31(d) optimisation of the reactants and transition states, but exact energies were not considered for simplicity.

{|
|-
|[[File:MO q2 endo cyy113.PNG|centre|frame|<center>Formation of Endo TS</center>]] ||[[File:MO q2 exo cyy113.PNG|centre|frame|<center>Formation of exo TS</center>]]
|-
| colspan="2" | <small><center>''Fig. 9: MO diagrams for the formation of the (A) endo TS and (B) exo TS''</center></small>
|}
{| class="wikitable"
|+'''Table 7: Relevant MOs in the endo and exo transition states'''
!
!Endo TS
!Exo TS
|-
|LUMO+1 (MO 43)
|
|
|-
|LUMO (MO 42)
|
|
|-
|HOMO (MO 41)
|
|
|-
|HOMO -1 (MO 40)
|
|
|}

Fig. 11 shows that secondary orbital interactions exist only for the endo TS. This has a stabilising effect as demonstrated in Fig. 12. Note that not all the orbitals are drawn; only those interacting are shown for simplicity.
[[File:Q2 secondary interaction cyy113.PNG|centre|frame|Fig. 12: Simplified MO diagram showing the secondary orbital interaction in the endo product]]
This secondary orbital interaction explains the ''endo rule'', which states that the endo product is always preferentially formed in a Diels-Alder reaction.<ref>J. García, J. Mayoral and L. Salvatella, ''European Journal of Organic Chemistry'', 2004, 2005, 85-90.</ref>
{| class="wikitable"
!Endo TS
!Exo TS
!Cheletropic
|-
|endo
|exo
|che
|}
{| class="wikitable"
|+'''Table 9: Energies of reactants, TS and products'''
! colspan="2" |Xylylene
! colspan="2" |SO<sub>2</sub>
! colspan="2" |Endo TS
! colspan="2" |Endo Product
|-
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
|-
|0.178746
|469.297659
|<nowiki>-0.118614</nowiki>
|<nowiki>-311.4210807</nowiki>
|0.090562
|237.770549
|0.021701
|56.9759798
|-
! colspan="2" |Exo TS
! colspan="2" |Exo Product
! colspan="2" |Cheletropic TS
! colspan="2" |Cheletropic Product
|-
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
!Energy (Hartrees)
!Energy (kJ mol<sup>-1</sup>)
|-
|0.092078
|241.750807
|0.021453
|56.3248558
|0.099062
|260.087301
|0.000005
|0.013127501
|}
{| class="wikitable"
|+'''Table 10: Reaction barriers and energies for the reaction between xylylene and SO<sub>2</sub>'''
!Reaction
!Formation of Endo Product
!Formation of Exo Product
!Formation of Cheletropic Product
|-
|Reaction barrier/ kJ mol<sup>-1</sup>
|<nowiki>+79.9</nowiki>
|<nowiki>+83.9</nowiki>
|<nowiki>+102.2</nowiki>
|-
|Reaction energy/ kJ mol<sup>-1</sup>
|<nowiki>-100.9</nowiki>
|<nowiki>-101.6</nowiki>
|<nowiki>-157.9</nowiki>
|}