ChemWiki - User contributions [en]

Rep:Mod:cej15 Transition States

2018-02-28T10:10:07Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph. When the gradient in the reaction profile reaches zero and the second derivative is negative, then that represents the transition state.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface<ref> Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448</ref>. The potential energy surface is a three-dimensional illustration of the reaction profile. A saddle point is when the gradient is zero but the second derivatives do not represent maxima or minima.

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sup>-1</sup> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sup>-1</sup> and -486.44 cm<sup>-1</sup> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T10:09:40Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph. When the gradient in the reaction profile reaches zero and the second derivative is negative, then that represents the transition state.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface<ref> Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448</ref>. The potential energy surface is a three-dimensional illustration of the reaction profile. A saddle point is when the gradient is zero but the second derivatives do not represent maxima or minima.

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sup>-1</sup> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T10:08:30Z

Cej15: /* Potential Energy Surface */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph. When the gradient in the reaction profile reaches zero and the second derivative is negative, then that represents the transition state.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface<ref> Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448</ref>. The potential energy surface is a three-dimensional illustration of the reaction profile. A saddle point is when the gradient is zero but the second derivatives do not represent maxima or minima.

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T10:08:07Z

Cej15: /* Potential Energy Surface */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph. When the gradient in the reaction profile reaches zero and the second derivative is negative, then that represents the transition state.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. <ref> Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448</ref>
The potential energy surface is a three-dimensional illustration of the reaction profile. A saddle point is when the gradient is zero but the second derivatives do not represent maxima or minima.

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T10:06:48Z

Cej15: /* Introduction */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph. When the gradient in the reaction profile reaches zero and the second derivative is negative, then that represents the transition state.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]
The potential energy surface is a three-dimensional illustration of the reaction profile. A saddle point is when the gradient is zero but the second derivatives do not represent maxima or minima.

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T09:58:08Z

Cej15: /* Transition State */

==Introduction==
===Transition State===
[[File:TS_cej.png|thumb|left|400px]]
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents, as shown in the graph.

===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

File:TS cej.png

2018-02-28T09:57:06Z

Cej15:

Rep:Mod:cej15 Transition States

2018-02-28T09:54:21Z

Cej15: /* Energies and Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|centre|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T09:53:57Z

Cej15: /* Energies and Analysis */

Rep:Mod:cej15 Transition States

2018-02-28T09:53:22Z

Cej15: /* Energies and Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

[[File:Reaction_profile_cej.png|thumb|600px]]

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

File:Reaction profile cej.png

2018-02-28T09:52:48Z

Cej15:

Rep:Mod:cej15 Transition States

2018-02-28T09:42:22Z

Cej15: /* MO Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

Shown below are the MO digrams of both the endo and exo transition states and the MO orbitals obtained from Gaussview.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo MO diagram || Exo MO diagram
|-
| [[File:ENDO_MO_cej.png|thumb|500px]] || [[File:EXO_MO_cej.png|thumb|500px]]
|}

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

From comparing the transition state HOMOs for the exo and endo adducts, it can be suggested that the endo adduct is kinetically favoured. This is because there are non-bonding interactions present between the reactive site of cyclohexadiene and the p-orbitals of the oxygen atoms on the 1,3-dioxole. This would stabilise the transition state and reduce its energy.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T09:34:54Z

Cej15: /* MO Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:Exercise_1_MO_cej.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}

===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

File:Exercise 1 MO cej.png

2018-02-28T09:34:02Z

Cej15:

File:EXO MO cej.png

2018-02-28T09:33:43Z

Cej15:

File:ENDO MO cej.png

2018-02-28T09:33:28Z

Cej15:

Rep:Mod:cej15 Transition States

2018-02-28T09:23:23Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Diels-Alder || Cheletropic
|-
| [[File:Exercise_3_DA_irc_cej.gif]] || [[File:Exercise_3_Chele_irc_cej.gif]]
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T09:22:28Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Endo Adduct || Exo Adduct
|-
| [[File:Exercise_2_endo_irc_cej.gif]] || [[File:Exercise_2_exo_irc_cej.gif]]
|}

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T09:21:24Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65 cm<sub>-1</sub> and the gifs for the vibrations and IRC are shown below.

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Vibrations || IRC
|-
| [[File:Exercise_1_vibration_cej.gif]] || [[File:Exercise_1_irc_cej.gif]]
|}

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

File:Exercise 3 DA irc cej.gif

2018-02-28T09:19:04Z

Cej15:

File:Exercise 3 Chele irc cej.gif

2018-02-28T09:18:48Z

Cej15:

File:Exercise 2 exo irc cej.gif

2018-02-28T09:18:34Z

Cej15:

File:Exercise 2 endo irc cej.gif

2018-02-28T09:18:20Z

Cej15:

File:Exercise 1 vibration cej.gif

2018-02-28T09:18:05Z

Cej15:

File:Exercise 1 irc cej.gif

2018-02-28T09:17:43Z

Cej15:

Rep:Mod:cej15 Transition States

2018-02-28T09:04:30Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
The transition states for the Diels-Alder reaction and the Cheletropic product were both determined and optimised and had vibrational frequencies of -351.62 cm<sub>-1</sub> and -486.44 cm<sub>-1</sub> respectively. Shown below are the intrinsic reaction coordinates of the two reactions.

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T08:59:37Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
The same procedure was performed for this reaction as that of exercise 1. The vibration frequencies obtained for the endo adduct and the exo adduct are -935.85 cm<sub>-1</sub> and -959.61 cm<sub>-1</sub>. The intrinsic reaction coordinates are shown below.

===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T08:52:30Z

Cej15: /* Optimisation and Determination of Transition State */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -948.65cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T08:05:53Z

Cej15: /* Energies and Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -949.59cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -853.60
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right. The same method of calculation was used as that of the previous exercise and the results are shown in the table below.

{| class="wikitable floatleft" style="text-align: center" border=1
|+ '''Change in Free Energies and Calculation of Activation Energies (kJmol<sup>-1</sup>)'''
! !! <math>\Delta_rG^{\ominus}(298.15K)</math> || <math>\Delta_rG^{\ddagger}</math>
|-
| Diels-Alder Product || + 4.50 || +0.01
|-
| Cheletropic Product || +4.57 || + 0.03
|}

The differences between the two are also very small but it could still be seen that the Diels-Alder product was more favourable as the change in free energy is more negative than that of the cheletropic product. The general energy profiles of the two can be shown in the graph below.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T07:48:47Z

Cej15: /* Energies and Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -949.59cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

The activation energy was also calculated by using the energies for the reactants and the energies at the transition state. Results for the activation energy of the endo and exo adducts are shown below respectively:

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

<div align="center"><math>\Delta_rG^{\ddagger}(298.15K) = -500.33 - (-233.32 + -267.07) = + 0.06 kJ/mol</math></div>

As the activation energy values for both are the same at 2 decimal places, both transition states are very similar in stability.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -fesfesfefesfesfse
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Liquid Simulations with milandeleev

2018-02-28T05:12:23Z

Cej15: /* Dynamical Properties Tasks */

== Introductory Tasks ==
=== Velocity Verlet Algorithm ===
A completed spreadsheet containing a model of this algorithm for a simple harmonic oscillator may be found here. The position of local maximum error with respect to time can be modelled by the following equation:

<div style="text-align: center;"><math>\max(E)=(5t-1)\cdot 8\times 10^{-5} </math></div>

In order to ensure that the total energy does not change by more than 1%, the timestep used must deceed 0.3 s. This lack of deviation is very important, as it ensures that the system obeys the law of conservation of energy, which means that it behaves in a physically consistent manner. In terms of the simple harmonic oscillator, then, it ensures that the observed wave frequency and period is constant due to the below equation:

<div style="text-align: center;"><math>E=h\nu</math></div>

=== Atomic Forces and Derivatives ===
A single Lennard-Jones potential reaches zero when the internuclear separation <math display="inline">r_0=\sigma</math>. This can be calculated by following the below process:

<div style="text-align: center;"><math>\phi(r)=4\epsilon\left ( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6}\right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{\sigma^{12}}{r_0^{12}}=\frac{\sigma^6}{r_0^6}</math></div><br />

<div style="text-align: center;"><math>r_0^{12}\sigma^6=r_0^6\sigma^{12}</math></div><br />

<div style="text-align: center;"><math>r_0=\sigma</math></div><br />

To calculate the force at this separation, it must be known that the force is the negative derivative of the energy with respect to the internuclear separation. Hence:

<div style="text-align: center;"><math> F = - \frac{\mathrm{d}\phi}{\mathrm{d}r}</math></div><br />

<div style="text-align: center;"><math>-\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r^7}\left (\frac{2\sigma^6}{r^6}-1 \right )</math></div><br />

Substituting in <math display="inline">r_0=\sigma</math>, then:

<div style="text-align: center;"><math>F_0=\frac{24\epsilon\sigma^6}{\sigma^7}\left (\frac{2\sigma^6}{\sigma^6}-1 \right )</math></div><br />

<div style="text-align: center;"><math>F_0=\frac{24\epsilon}{\sigma}</math></div><br />

To calculate the equilibrium separation <math display="inline">r_{eq}</math>, the minimum point of the energy curve must be found. This can be achieved by differentiating the equation and equating it to zero, and solving for <math display="inline">r</math>, which is equivalent to finding the location whereat <math display="inline">F=0</math>.

<div style="text-align: center;"><math>\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r_{eq}^7}\left (1-\frac{2\sigma^6}{r_{eq}^6} \right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{2\sigma^6}{r_{eq}^6}=1</math></div><br />

<div style="text-align: center;"><math>2\sigma^6=r_{eq}^6</math></div><br />

<div style="text-align: center;"><math>r_{eq}=2^{\frac{1}{6}}\sigma</math></div><br />

The well depth <math display="inline">\phi \left ( r_{eq} \right )</math> thereof:

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{r_{eq}^{12}} - \frac{\sigma^6}{r_{eq}^6}\right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{ \left (2^{\frac{1}{6}}\sigma \right )^{12}} - \frac{\sigma^6}{\left ( 2^{\frac{1}{6}}\sigma \right )^6} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{1}{4} - \frac{1}{2} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = \epsilon</math></div><br />

=== Atomic Forces and Integrals ===
The integral below is a generalised form to which boundary conditions can be applied.

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=\int\limits_{L_1}^{L_2} 4\epsilon\left ( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6}\right )\mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r) =4\epsilon\sigma^{12} \int\limits_{L_1}^{L_2} r^{-12} \mathrm{d}r - 4\epsilon\sigma^{6}\int\limits_{L_1}^{L_2} r^{-6} \mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=4\epsilon\sigma^{6} \left [ \frac {1}{5r^5} - \frac{\sigma^{6}}{11r^{11}} \right ]_{L_1}^{L_2} </math></div><br />

The limits <math display=inline>L_1=\{2\sigma, 2.5\sigma, 3\sigma\}</math> and <math display=inline>L_2=\infty</math> can be applied as below. The final results apply for <math display=inline>\sigma=\epsilon=1</math>.

<div style="text-align: center;"><math>L_2=\infty \therefore \left ( \frac {1}{5(\infty)^5} - \frac{\sigma^{6}}{11(\infty)^{11}} \right ) = 0 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {\sigma^6}{11L_1^{11}} - \frac {1}{5L_1^5} \right ) = 4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11L_1^6}{55L_1^{11}} \right ) </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2\sigma)^6}{55(2\sigma)^{11}} \right ) = -\frac{699\sigma \epsilon}{28160} = -0.0248 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2.5\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2.5\sigma)^6}{55(2.5\sigma)^{11}} \right ) = -\frac{4391808\sigma \epsilon}{537109375} = -0.00818 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{3\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {3\sigma^6-11(2.5\sigma)^6}{55(3\sigma)^{11}} \right ) = -\frac{32056\sigma \epsilon}{9743085} = -0.00329 </math></div><br />

=== Periodic Boundary Conditions ===
1 mL of water at STP has a mass of 1 g. Consequently, the number of moles of water in such a volume ''n'' and the number of molecules ''N'' is calculated below. The final result is ''N'' = 3.43 × 10<sup>23</sup> molecules. <br />.

<div style="text-align: center;"><math>n = \frac{m}{M_r} = \frac{1}{18.01528} = 0.0555084350618 </math></div><br />

<div style="text-align: center;"><math>N = n \cdot N_A = 0.0555084350618 \cdot 6.022140857 \times 10^{23} \approxeq 3.343 \times 10^{22} </math></div><br />

The calculation for the volume of 10000 molecules of water at STP is below. The final result is 2.99 × 10<sup>-19</sup> mL. <br />

<div style="text-align: center;"><math>n = \frac{N}{N_A} = \frac{10000}{6.022140857 \times 10^{23}} = 1.6605390404272 \times 10^-20 </math></div><br />

<div style="text-align: center;"><math>V = m = n\cdot M_r = 1.6605390404272 \times 10^{-20} \cdot 18.01528 \approxeq 2.992 \times 10^{-19} </math></div><br />

If an atom at (0.5, 0.5, 0.5) in a unit cube with repetitive boundary conditions (like the game Snake) moves along a vector of (0.7, 0.6, 0.2), it will reappear in the cube at (0.2, 0.1, 0.7).

=== Reduced Units ===
For argon, the Lennard-Jones parameters are ''σ'' = 0.34 nm and ''ϵ'' = 120''k''<sub>''B''</sub>.

<div style="text-align: center;"><math>r = r^* \cdot \sigma = 3.2 \cdot 0.34 = 1.088 \text{ nm} </math></div><br />

<div style="text-align: center;"><math>\epsilon = 120k_B = 1.656778224 \times 10^{-21} J \approxeq -2.751 \times 10^{-48} \text { kJ/mol} </math></div><br />

<div style="text-align: center;"><math>T=T^*\frac{\epsilon}{k_B} = 1.5 \cdot 120 = 180\text{ K} </math></div><br />

== Equilibration Tasks ==
=== Creating the Simulation Box ===
If atoms are generated with random coordinates, one atom could be generated in a position that overlaps with another atom. This would dramatically increase the overall energy due to intra-pair repulsive potentials, destabilising the system.

A relative lattice spacing of 1.07722 for a simple cubic lattice (1 lattice point per unit cell) generates a lattice point per unit volume density of 0.79999. Generating an atom at each lattice point for a 10×10×10 box produces 1000 atoms. A relative lattice spacing of 1.49380 for a face-centered cubic lattice (4 lattice points per unit cell) generates a lattice point per unit volume density of 1.2. Generating an atom at each lattice point for a 10×10×10 box produces 4000 atoms.

=== Setting the Properties of the Atoms ===
The following commands in LAMMPS have specific functions, as described below:

<pre>mass 1 1.0</pre> This command creates a type 1 atom with mass 1.

<pre>pair_style lj/cut 3.0</pre> This command instructs the program to compute Leonard-Jones pairwise potentials, using a cutoff point of 3.

<pre>pair_coeff * * 1.0 1.0</pre> This command gives the force field coefficients (''ϵ'' and ''σ'') between two type 1 atoms.

As the initial positions and velocities have been specified, this is consistent with the Velocity-Verlet algorithm.

=== Running the Simulation ===
Using piece of code that references the input value means that the rest of the code need not be altered when the input is altered. Furthermore, other values that depend on the given timestep can also be linked straight back to the input value.

=== Checking Equilibration ===
The simulation data of total particular energy, temperature and pressure against time for a 0.001 s timestep are graphed below (in that order). The simulation reaches a clear equilibrium for all three values, as, after a short period of time (ca. 0.03 s, or 30 timesteps).

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | Simulation Data
|-
| [[File:Mk energytime1.png|450px|center]] || [[File:Mk pressuretime1.png|450px|center]] || [[File:Mk temperaturetime1.png|450px|center]]
|}

The data for different timesteps is graphed below. It is clear that the data for the 0.001 s and 0.0025 s are very similar and can essentially be used interchangeably. In the case of simulations over long periods of time, the 0.0025 s timestep is appropriate to reduce computation time but still maintain a high level of accuracy (the mean energy value differs by 0.005%). The 0.015 s timestep does not reach an average energy value, but instead continually increases the total particular energy with time: indicative of an unstable, positively feeding-back system. The Velocity-Verlet algorithm requires the initial atomic positions and velocities, and then simulates molecular movement from there. A large timestep results in large atomic displacements per timestep, causing a significant error in the calculation which multiplies by each timestep.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of Total Particular Energies for Different Timesteps
|-
| [[File:Mk timesteptime1.png|700px|center]]
|}

== Specific Condition Simulations Tasks ==

=== Thermostats and Barostats ===
Setting γ as the velocity scaling factor to inter-convert between the instantaneous and target temperatures allows elucidation of its value in terms of the latter.

<div align="center"><math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math></div><br/>

<div align="center"><math>\frac{1}{2}\sum_i m_i \gamma^2 v_i^2 = \frac{3}{2} N k_B \mathfrak{T}</math></div><br/>

<div align="center"><math>\frac{\frac{1}{2}\sum_i m_i v_i^2}{\frac{1}{2}\sum_i m_i \gamma^2 v_i^2} = \frac{\frac{3}{2} N k_B T}{\frac{3}{2} N k_B \mathfrak{T}}</math></div><br/>

<div align="center"><math>\frac{1}{\gamma^2} = \frac{T}{\mathfrak{T}}</math></div><br/>

<div align="center"><math> \gamma = \pm \sqrt{\frac{\mathfrak{T}}{T}}</math></div><br/>

=== Examining the Input Script ===
In the program input scripts for the simulations in this section, there exists a line of code that resembles that below:

<pre> fix aves all ave/time 100 1000 100000 v_dens v_temp v_press v_dens2 v_temp2 v_press2 </pre>

The first numerical variable (100 in this case) represents the number of timesteps τ<sub>i</sub> over which an average is taken. Mathematically:

<div align="center"><math>\bar{v} _n = \frac {\sum_i \tau_i}{100}</math></div><br/>

The second numerical variable (1000 in this case) represents the number of averages over which to take a further average. Mathematically:

<div align="center"><math>\tilde{v} _i = \frac {\sum_n \bar{v} _n}{1000}</math></div><br/>

The third numerical variable (10000 in this case) represents the number of previous averages over which to take the final average. Mathematically:

<div align="center"><math>\hat{v} _a = \frac {\sum_i \tilde{v} _i}{10000}</math></div><br/>

=== Plotting the Equations of State ===

<div align="center"><math>\rho = \frac {p}{T}</math></div><br/>
The data generated from the simulations in this section have been plotted in terms of temperature and density at reduced pressures 2 and 3 in the graph below (with error bars included). The ideal density calculated by the ideal gas law in reduced units (as above) is also plotted. The graph shows a clear discrepancy between the predicted and simulated densities, which is due to the ideal gas law treating the atoms as classical hard balls that experience no repulsion unless they are in direct contact (bouncing off each other). However, the Lennard-Jones potential models the atoms more faithfully to reality, in that, within a critical radius, the electrons orbiting the atoms repel each other and the atoms experience net repulsion. The densities predicted by the ideal gas are consequently higher as they do not take neighbour repulsions into consideration, which means that the model predicts that the atoms can be within closer proximities of one another without destabilising the system.

However, it is also clear that the discrepancy decreases with increasing temperature. This is because, at higher temperatures, the atoms have more thermal energy and thus behave more like Newtonian hard balls, as their kinetic energy is dominant over the repulsive electronic forces until a smaller distance. Essentially, the atoms can get closer as their thermal energy can overcome the electronic repulsion.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Computed and Ideal Densities at varying Temperatures and Pressures
|-
| [[File:Mk temperaturedensity1.png|700px|center]]
|}

== Statistical Physics Tasks ==

In this section, ten simulations were again run to determine the change in heat capacity per unit volume for a liquid at varying densities. An exemplar script can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Script_examples_with_milandeleev here]. The results are graphed below. Higher density very obviously produces a higher heat capacity, which is expected because more atoms in one space can absorb more heat energy than fewer atoms in the same space. There is a clear decrease in the heat capacity with increasing temperature, which is at odds with the physical predictions. At lower temperatures, there are fewer thermally accessible translational, rotational, electronic and vibrational energy levels available. As temperature increases, more of these states become available so more of the energy levels become populated, making the internal energy rise rapidly. Since the heat capacity is the derivative of the internal energy with respect to temperature, higher temperatures are thus expected to increase the heat capacity (although this does level out towards the phase transition temperature). This deviation from the expected trend could possibly be explained by the fact that this system is modelling simple hydrogen atom fluids, and thus rotational and vibrational energy levels are not available. Furthermore the software itself may not take into account electronic transitions.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of the Variation of Heat Capacities per Unit Volume with Temperature for Varying Densities
|-
| [[File:Mk temperatureheatcap1.png|700px|center]]
|}

== Radial Distribution Function Tasks ==

The radial distribution functions of multiple phases and their integrals are plotted against distance below. The radial distribution function for these simulations should average out to one, irrespective of the phase, because the function itself is a measure of the particle density surrounding a given particle. Consequently it can be seen as a measure of how consistently structured a material is: for a gas, the position of the particles is completely unpredictable and so it will very quickly tend to a central value. However, for a crystalline solid, the peaks will average unity but will never become a straight line that tend to unity, because the crystal has a defined structure with particles a fixed distance away from each other. Henceforth, there is a much greater probability that a particle will be found in a lattice site than outside of one, so the RDF will never lose its 'bumpiness'. From this line of reasoning, the RDF of a simple liquid should tend to unity but slower than the same function for a gas.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | RDF and Integrated RDF for Three Phases
|-
| [[File:Mk rdfr1.png|450px|center]] || [[File:Mk intrdfr1.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

To find the coordination number, the value of the first minimum on the RDF of the solid is found (which appears at <math display="inline">r=1.975</math>), which equates to a coordination number of 12. The lattice spacing, consequently, is ca. 1.37.

== Dynamical Properties Tasks ==

The mean squared displacements for a small sample and a large sample in different phases are plotted below.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | Mean Squared Displacement for Varying Phases and Sample Sizes
|-
| Fewer Atoms || One Million Atoms
|-
| [[File:Mk msdt1.png|450px|center]] || [[File:Mk msdt2.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

The resulting diffusion coefficients, calculated by the equation below (in which ''n'' is the dimensionality, or 3 in this case), are also tabulated below.

<div align="center"><math> \frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}=2nD</math></div>

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | <math>D</math> (m<sup>2</sup>s<sup>-1</sup>) to 8 s.f
|-
| Phase || Small Sample || Large Sample
|-
| Solid || 0.010172398 || 5.4978008 × 10<sup>-7</sup>
|-
| Liquid || 0.19168932 || 0.088731507
|-
| Gas || 2.7676196 || 3.0690123
|}

In general, these findings are logically consistent. The mean-squared displacement and thus the diffusion coefficient should be highest for a gas, as the particles have a higher thermal energy and so move away from their original positions fastest. Liquids have lower diffusion coefficients, and solids lower still. However, comparing the simulation for fewer and more atoms reveals some odd data. The diffusion coefficients for the gas phase are relatively similar, with a small increase on the side of the larger sample. This difference in value can simply be attributed to the greater accuracy of the million-atoms calculation. However, the value of <math display=inline>D</math> for the liquid simulation is smaller for one million atoms, and for a solid it is much smaller still (by a magnitude of one hundred thousand). The values should at least be similar, as the parameters are similar, but the sample sizes are different. To rationalise this, it is possible to consider that the value for <math display=inline>D</math> for one million atoms as a solid is skewed due to the presence of negative micro-gradient values, so it is difficult to compare in any valid sense to the computation for fewer atoms. For the case of the liquid, the origin of the difference is difficult to reason.

== Academic Report ==

=== Abstract ===
100 Words. In the abstract, you should provide an overview of your results so a quick (and naive at this point) data based conclusion can be made on the original hypothesis. It's a section you write at the end and doesn't go into as much detail as the conclusion but you sum up the relevance of the research (intro), what you have done, what results do you have to show that you have done this and the ultimate conclusion. Abstract - what do you want to fundamentally do, what have you done, how does this support the original idea.

=== Introduction ===
100 Words. Your introduction paints a picture of the background of the research; what has been done by others and where is the niche for your work. How will your work benefit the community - maybe it's a new technique. You open up the niche so the subsequent discussion inserts your work into the existing community or (for high impact journals like Nature) creates a question and novel direction that can be picked up and worked on by other interested parties. In this lab, it would be good if you can demonstrate some of the importance of your results and impact it had on Science and society.

=== Aims and Objectives ===
80 Words. We spend an awful lot of time in this lab changing properties and measuring their physical outputs. I think fundamentally, we explore phase in this lab and the definition of phase.

=== Methodology ===
320 Words. You are right, we spend an awful lot of time in this lab changing properties and measuring their physical outputs. I think fundamentally, we explore phase in this lab and the definition of phase. Reproducibility of data reinforces the high standards and quality expected for a piece of work in a peer reviewed journal. It also engenders the work with legitimacy. It is not true that the velocity-Verlet requires the previous position as an input - it requires the starting positions as input parameters and works it's way from there. We usually do not talk about timestep when writing about computational Chemistry because hopefully we have picked the timestep that simulates the system accurately for the least amount of time. For someone who did not know which software you used, it might be an idea to mention this here. In terms of syntax specific for lammps (pair_style, pair_coeff) this should be discussed in terms of interaction strengths and forcefields rather than the specific syntax.

=== Results and Discussion ===
600 Words. Present a result using a method described and discuss what this result means and how your results show it.

=== Conclusion ===
100 Words. Tie up the research and what your results show. Reread your intro, what are you trying to do? How does your research do this?

Rep:Liquid Simulations with milandeleev

2018-02-28T05:02:12Z

Cej15: /* Dynamical Properties Tasks */

Rep:Liquid Simulations with milandeleev

2018-02-28T05:01:46Z

Cej15: /* Plotting the Equations of State */

== Introductory Tasks ==
=== Velocity Verlet Algorithm ===
A completed spreadsheet containing a model of this algorithm for a simple harmonic oscillator may be found here. The position of local maximum error with respect to time can be modelled by the following equation:

<div style="text-align: center;"><math>\max(E)=(5t-1)\cdot 8\times 10^{-5} </math></div>

In order to ensure that the total energy does not change by more than 1%, the timestep used must deceed 0.3 s. This lack of deviation is very important, as it ensures that the system obeys the law of conservation of energy, which means that it behaves in a physically consistent manner. In terms of the simple harmonic oscillator, then, it ensures that the observed wave frequency and period is constant due to the below equation:

<div style="text-align: center;"><math>E=h\nu</math></div>

=== Atomic Forces and Derivatives ===
A single Lennard-Jones potential reaches zero when the internuclear separation <math display="inline">r_0=\sigma</math>. This can be calculated by following the below process:

<div style="text-align: center;"><math>\phi(r)=4\epsilon\left ( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6}\right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{\sigma^{12}}{r_0^{12}}=\frac{\sigma^6}{r_0^6}</math></div><br />

<div style="text-align: center;"><math>r_0^{12}\sigma^6=r_0^6\sigma^{12}</math></div><br />

<div style="text-align: center;"><math>r_0=\sigma</math></div><br />

To calculate the force at this separation, it must be known that the force is the negative derivative of the energy with respect to the internuclear separation. Hence:

<div style="text-align: center;"><math> F = - \frac{\mathrm{d}\phi}{\mathrm{d}r}</math></div><br />

<div style="text-align: center;"><math>-\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r^7}\left (\frac{2\sigma^6}{r^6}-1 \right )</math></div><br />

Substituting in <math display="inline">r_0=\sigma</math>, then:

<div style="text-align: center;"><math>F_0=\frac{24\epsilon\sigma^6}{\sigma^7}\left (\frac{2\sigma^6}{\sigma^6}-1 \right )</math></div><br />

<div style="text-align: center;"><math>F_0=\frac{24\epsilon}{\sigma}</math></div><br />

To calculate the equilibrium separation <math display="inline">r_{eq}</math>, the minimum point of the energy curve must be found. This can be achieved by differentiating the equation and equating it to zero, and solving for <math display="inline">r</math>, which is equivalent to finding the location whereat <math display="inline">F=0</math>.

<div style="text-align: center;"><math>\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r_{eq}^7}\left (1-\frac{2\sigma^6}{r_{eq}^6} \right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{2\sigma^6}{r_{eq}^6}=1</math></div><br />

<div style="text-align: center;"><math>2\sigma^6=r_{eq}^6</math></div><br />

<div style="text-align: center;"><math>r_{eq}=2^{\frac{1}{6}}\sigma</math></div><br />

The well depth <math display="inline">\phi \left ( r_{eq} \right )</math> thereof:

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{r_{eq}^{12}} - \frac{\sigma^6}{r_{eq}^6}\right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{ \left (2^{\frac{1}{6}}\sigma \right )^{12}} - \frac{\sigma^6}{\left ( 2^{\frac{1}{6}}\sigma \right )^6} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{1}{4} - \frac{1}{2} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = \epsilon</math></div><br />

=== Atomic Forces and Integrals ===
The integral below is a generalised form to which boundary conditions can be applied.

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=\int\limits_{L_1}^{L_2} 4\epsilon\left ( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6}\right )\mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r) =4\epsilon\sigma^{12} \int\limits_{L_1}^{L_2} r^{-12} \mathrm{d}r - 4\epsilon\sigma^{6}\int\limits_{L_1}^{L_2} r^{-6} \mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=4\epsilon\sigma^{6} \left [ \frac {1}{5r^5} - \frac{\sigma^{6}}{11r^{11}} \right ]_{L_1}^{L_2} </math></div><br />

The limits <math display=inline>L_1=\{2\sigma, 2.5\sigma, 3\sigma\}</math> and <math display=inline>L_2=\infty</math> can be applied as below. The final results apply for <math display=inline>\sigma=\epsilon=1</math>.

<div style="text-align: center;"><math>L_2=\infty \therefore \left ( \frac {1}{5(\infty)^5} - \frac{\sigma^{6}}{11(\infty)^{11}} \right ) = 0 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {\sigma^6}{11L_1^{11}} - \frac {1}{5L_1^5} \right ) = 4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11L_1^6}{55L_1^{11}} \right ) </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2\sigma)^6}{55(2\sigma)^{11}} \right ) = -\frac{699\sigma \epsilon}{28160} = -0.0248 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2.5\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2.5\sigma)^6}{55(2.5\sigma)^{11}} \right ) = -\frac{4391808\sigma \epsilon}{537109375} = -0.00818 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{3\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {3\sigma^6-11(2.5\sigma)^6}{55(3\sigma)^{11}} \right ) = -\frac{32056\sigma \epsilon}{9743085} = -0.00329 </math></div><br />

=== Periodic Boundary Conditions ===
1 mL of water at STP has a mass of 1 g. Consequently, the number of moles of water in such a volume ''n'' and the number of molecules ''N'' is calculated below. The final result is ''N'' = 3.43 × 10<sup>23</sup> molecules. <br />.

<div style="text-align: center;"><math>n = \frac{m}{M_r} = \frac{1}{18.01528} = 0.0555084350618 </math></div><br />

<div style="text-align: center;"><math>N = n \cdot N_A = 0.0555084350618 \cdot 6.022140857 \times 10^{23} \approxeq 3.343 \times 10^{22} </math></div><br />

The calculation for the volume of 10000 molecules of water at STP is below. The final result is 2.99 × 10<sup>-19</sup> mL. <br />

<div style="text-align: center;"><math>n = \frac{N}{N_A} = \frac{10000}{6.022140857 \times 10^{23}} = 1.6605390404272 \times 10^-20 </math></div><br />

<div style="text-align: center;"><math>V = m = n\cdot M_r = 1.6605390404272 \times 10^{-20} \cdot 18.01528 \approxeq 2.992 \times 10^{-19} </math></div><br />

If an atom at (0.5, 0.5, 0.5) in a unit cube with repetitive boundary conditions (like the game Snake) moves along a vector of (0.7, 0.6, 0.2), it will reappear in the cube at (0.2, 0.1, 0.7).

=== Reduced Units ===
For argon, the Lennard-Jones parameters are ''σ'' = 0.34 nm and ''ϵ'' = 120''k''<sub>''B''</sub>.

<div style="text-align: center;"><math>r = r^* \cdot \sigma = 3.2 \cdot 0.34 = 1.088 \text{ nm} </math></div><br />

<div style="text-align: center;"><math>\epsilon = 120k_B = 1.656778224 \times 10^{-21} J \approxeq -2.751 \times 10^{-48} \text { kJ/mol} </math></div><br />

<div style="text-align: center;"><math>T=T^*\frac{\epsilon}{k_B} = 1.5 \cdot 120 = 180\text{ K} </math></div><br />

== Equilibration Tasks ==
=== Creating the Simulation Box ===
If atoms are generated with random coordinates, one atom could be generated in a position that overlaps with another atom. This would dramatically increase the overall energy due to intra-pair repulsive potentials, destabilising the system.

A relative lattice spacing of 1.07722 for a simple cubic lattice (1 lattice point per unit cell) generates a lattice point per unit volume density of 0.79999. Generating an atom at each lattice point for a 10×10×10 box produces 1000 atoms. A relative lattice spacing of 1.49380 for a face-centered cubic lattice (4 lattice points per unit cell) generates a lattice point per unit volume density of 1.2. Generating an atom at each lattice point for a 10×10×10 box produces 4000 atoms.

=== Setting the Properties of the Atoms ===
The following commands in LAMMPS have specific functions, as described below:

<pre>mass 1 1.0</pre> This command creates a type 1 atom with mass 1.

<pre>pair_style lj/cut 3.0</pre> This command instructs the program to compute Leonard-Jones pairwise potentials, using a cutoff point of 3.

<pre>pair_coeff * * 1.0 1.0</pre> This command gives the force field coefficients (''ϵ'' and ''σ'') between two type 1 atoms.

As the initial positions and velocities have been specified, this is consistent with the Velocity-Verlet algorithm.

=== Running the Simulation ===
Using piece of code that references the input value means that the rest of the code need not be altered when the input is altered. Furthermore, other values that depend on the given timestep can also be linked straight back to the input value.

=== Checking Equilibration ===
The simulation data of total particular energy, temperature and pressure against time for a 0.001 s timestep are graphed below (in that order). The simulation reaches a clear equilibrium for all three values, as, after a short period of time (ca. 0.03 s, or 30 timesteps).

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | Simulation Data
|-
| [[File:Mk energytime1.png|450px|center]] || [[File:Mk pressuretime1.png|450px|center]] || [[File:Mk temperaturetime1.png|450px|center]]
|}

The data for different timesteps is graphed below. It is clear that the data for the 0.001 s and 0.0025 s are very similar and can essentially be used interchangeably. In the case of simulations over long periods of time, the 0.0025 s timestep is appropriate to reduce computation time but still maintain a high level of accuracy (the mean energy value differs by 0.005%). The 0.015 s timestep does not reach an average energy value, but instead continually increases the total particular energy with time: indicative of an unstable, positively feeding-back system. The Velocity-Verlet algorithm requires the initial atomic positions and velocities, and then simulates molecular movement from there. A large timestep results in large atomic displacements per timestep, causing a significant error in the calculation which multiplies by each timestep.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of Total Particular Energies for Different Timesteps
|-
| [[File:Mk timesteptime1.png|700px|center]]
|}

== Specific Condition Simulations Tasks ==

=== Thermostats and Barostats ===
Setting γ as the velocity scaling factor to inter-convert between the instantaneous and target temperatures allows elucidation of its value in terms of the latter.

<div align="center"><math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math></div><br/>

<div align="center"><math>\frac{1}{2}\sum_i m_i \gamma^2 v_i^2 = \frac{3}{2} N k_B \mathfrak{T}</math></div><br/>

<div align="center"><math>\frac{\frac{1}{2}\sum_i m_i v_i^2}{\frac{1}{2}\sum_i m_i \gamma^2 v_i^2} = \frac{\frac{3}{2} N k_B T}{\frac{3}{2} N k_B \mathfrak{T}}</math></div><br/>

<div align="center"><math>\frac{1}{\gamma^2} = \frac{T}{\mathfrak{T}}</math></div><br/>

<div align="center"><math> \gamma = \pm \sqrt{\frac{\mathfrak{T}}{T}}</math></div><br/>

=== Examining the Input Script ===
In the program input scripts for the simulations in this section, there exists a line of code that resembles that below:

<pre> fix aves all ave/time 100 1000 100000 v_dens v_temp v_press v_dens2 v_temp2 v_press2 </pre>

The first numerical variable (100 in this case) represents the number of timesteps τ<sub>i</sub> over which an average is taken. Mathematically:

<div align="center"><math>\bar{v} _n = \frac {\sum_i \tau_i}{100}</math></div><br/>

The second numerical variable (1000 in this case) represents the number of averages over which to take a further average. Mathematically:

<div align="center"><math>\tilde{v} _i = \frac {\sum_n \bar{v} _n}{1000}</math></div><br/>

The third numerical variable (10000 in this case) represents the number of previous averages over which to take the final average. Mathematically:

<div align="center"><math>\hat{v} _a = \frac {\sum_i \tilde{v} _i}{10000}</math></div><br/>

=== Plotting the Equations of State ===

<div align="center"><math>\rho = \frac {p}{T}</math></div><br/>
The data generated from the simulations in this section have been plotted in terms of temperature and density at reduced pressures 2 and 3 in the graph below (with error bars included). The ideal density calculated by the ideal gas law in reduced units (as above) is also plotted. The graph shows a clear discrepancy between the predicted and simulated densities, which is due to the ideal gas law treating the atoms as classical hard balls that experience no repulsion unless they are in direct contact (bouncing off each other). However, the Lennard-Jones potential models the atoms more faithfully to reality, in that, within a critical radius, the electrons orbiting the atoms repel each other and the atoms experience net repulsion. The densities predicted by the ideal gas are consequently higher as they do not take neighbour repulsions into consideration, which means that the model predicts that the atoms can be within closer proximities of one another without destabilising the system.

However, it is also clear that the discrepancy decreases with increasing temperature. This is because, at higher temperatures, the atoms have more thermal energy and thus behave more like Newtonian hard balls, as their kinetic energy is dominant over the repulsive electronic forces until a smaller distance. Essentially, the atoms can get closer as their thermal energy can overcome the electronic repulsion.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Computed and Ideal Densities at varying Temperatures and Pressures
|-
| [[File:Mk temperaturedensity1.png|700px|center]]
|}

== Statistical Physics Tasks ==

In this section, ten simulations were again run to determine the change in heat capacity per unit volume for a liquid at varying densities. An exemplar script can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Script_examples_with_milandeleev here]. The results are graphed below. Higher density very obviously produces a higher heat capacity, which is expected because more atoms in one space can absorb more heat energy than fewer atoms in the same space. There is a clear decrease in the heat capacity with increasing temperature, which is at odds with the physical predictions. At lower temperatures, there are fewer thermally accessible translational, rotational, electronic and vibrational energy levels available. As temperature increases, more of these states become available so more of the energy levels become populated, making the internal energy rise rapidly. Since the heat capacity is the derivative of the internal energy with respect to temperature, higher temperatures are thus expected to increase the heat capacity (although this does level out towards the phase transition temperature). This deviation from the expected trend could possibly be explained by the fact that this system is modelling simple hydrogen atom fluids, and thus rotational and vibrational energy levels are not available. Furthermore the software itself may not take into account electronic transitions.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of the Variation of Heat Capacities per Unit Volume with Temperature for Varying Densities
|-
| [[File:Mk temperatureheatcap1.png|700px|center]]
|}

== Radial Distribution Function Tasks ==

The radial distribution functions of multiple phases and their integrals are plotted against distance below. The radial distribution function for these simulations should average out to one, irrespective of the phase, because the function itself is a measure of the particle density surrounding a given particle. Consequently it can be seen as a measure of how consistently structured a material is: for a gas, the position of the particles is completely unpredictable and so it will very quickly tend to a central value. However, for a crystalline solid, the peaks will average unity but will never become a straight line that tend to unity, because the crystal has a defined structure with particles a fixed distance away from each other. Henceforth, there is a much greater probability that a particle will be found in a lattice site than outside of one, so the RDF will never lose its 'bumpiness'. From this line of reasoning, the RDF of a simple liquid should tend to unity but slower than the same function for a gas.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | RDF and Integrated RDF for Three Phases
|-
| [[File:Mk rdfr1.png|450px|center]] || [[File:Mk intrdfr1.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

To find the coordination number, the value of the first minimum on the RDF of the solid is found (which appears at <math display="inline">r=1.975</math>), which equates to a coordination number of 12. The lattice spacing, consequently, is ca. 1.37.

== Dynamical Properties Tasks ==

The mean squared displacements for a small sample and a large sample in different phases are plotted below.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | Mean Squared Displacement for Varying Phases and Sample Sizes
|-
| Fewer Atoms || One Million Atoms
|-
| [[File:Mk msdt1.png|450px|center]] || [[File:Mk msdt2.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

The resulting diffusion coefficients, calculated by the equation below (in which ''n'' is the dimensionality, or 3 in this case), are also tabulated below.

<div align="center"><math> \frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}=2nD</math></div>

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | <math>D</math> (m<sup>2</sup>s<sup>-1</sup>) to 8 s.f
|-
| Phase || Small Sample || Large Sample
|-
| Solid || 0.010172398 || 5.4978008 × 10<sup>-7</sup>
|-
| Liquid || 0.19168932 || 0.088731507
|-
| Gas || 2.7676196 || 3.0690123
|}

In general, these findings are logically consistent. The mean-squared displacement and thus the diffusion coefficient should be highest for a gas, as the particles have a higher kinetic energy and so move away from their original positions fastest. Liquids have lower diffusion coefficients, and solids lower still. However, comparing the simulation for fewer and more atoms reveals some odd data. The diffusion coefficients for the gas phase are relatively similar, with a small increase on the side of the larger sample. This difference in value can simply be attributed to the greater accuracy of the million-atoms calculation. However, the value of <math display=inline>D</math> for the liquid simulation is smaller for one million atoms, and for a solid it is much smaller still (by a magnitude of one hundred thousand). The values should at least be similar, as the parameters are similar, but the sample sizes are different. To rationalise this, it is possible to consider that the value for <math display=inline>D</math> for one million atoms as a solid is skewed due to the presence of negative micro-gradient values, so it is difficult to compare in any valid sense to the computation for fewer atoms. For the case of the liquid, the origin of the difference is difficult to reason.

Rep:Liquid Simulations with milandeleev

2018-02-28T04:58:36Z

Cej15: /* Dynamical Properties Tasks */

== Introductory Tasks ==
=== Velocity Verlet Algorithm ===
A completed spreadsheet containing a model of this algorithm for a simple harmonic oscillator may be found here. The position of local maximum error with respect to time can be modelled by the following equation:

<div style="text-align: center;"><math>\max(E)=(5t-1)\cdot 8\times 10^{-5} </math></div>

In order to ensure that the total energy does not change by more than 1%, the timestep used must deceed 0.3 s. This lack of deviation is very important, as it ensures that the system obeys the law of conservation of energy, which means that it behaves in a physically consistent manner. In terms of the simple harmonic oscillator, then, it ensures that the observed wave frequency and period is constant due to the below equation:

<div style="text-align: center;"><math>E=h\nu</math></div>

=== Atomic Forces and Derivatives ===
A single Lennard-Jones potential reaches zero when the internuclear separation <math display="inline">r_0=\sigma</math>. This can be calculated by following the below process:

<div style="text-align: center;"><math>\phi(r)=4\epsilon\left ( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6}\right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{\sigma^{12}}{r_0^{12}}=\frac{\sigma^6}{r_0^6}</math></div><br />

<div style="text-align: center;"><math>r_0^{12}\sigma^6=r_0^6\sigma^{12}</math></div><br />

<div style="text-align: center;"><math>r_0=\sigma</math></div><br />

To calculate the force at this separation, it must be known that the force is the negative derivative of the energy with respect to the internuclear separation. Hence:

<div style="text-align: center;"><math> F = - \frac{\mathrm{d}\phi}{\mathrm{d}r}</math></div><br />

<div style="text-align: center;"><math>-\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r^7}\left (\frac{2\sigma^6}{r^6}-1 \right )</math></div><br />

Substituting in <math display="inline">r_0=\sigma</math>, then:

<div style="text-align: center;"><math>F_0=\frac{24\epsilon\sigma^6}{\sigma^7}\left (\frac{2\sigma^6}{\sigma^6}-1 \right )</math></div><br />

<div style="text-align: center;"><math>F_0=\frac{24\epsilon}{\sigma}</math></div><br />

To calculate the equilibrium separation <math display="inline">r_{eq}</math>, the minimum point of the energy curve must be found. This can be achieved by differentiating the equation and equating it to zero, and solving for <math display="inline">r</math>, which is equivalent to finding the location whereat <math display="inline">F=0</math>.

<div style="text-align: center;"><math>\frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{24\epsilon\sigma^6}{r_{eq}^7}\left (1-\frac{2\sigma^6}{r_{eq}^6} \right )=0</math></div><br />

<div style="text-align: center;"><math>\frac{2\sigma^6}{r_{eq}^6}=1</math></div><br />

<div style="text-align: center;"><math>2\sigma^6=r_{eq}^6</math></div><br />

<div style="text-align: center;"><math>r_{eq}=2^{\frac{1}{6}}\sigma</math></div><br />

The well depth <math display="inline">\phi \left ( r_{eq} \right )</math> thereof:

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{r_{eq}^{12}} - \frac{\sigma^6}{r_{eq}^6}\right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{\sigma^{12}}{ \left (2^{\frac{1}{6}}\sigma \right )^{12}} - \frac{\sigma^6}{\left ( 2^{\frac{1}{6}}\sigma \right )^6} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = 4\epsilon\left ( \frac{1}{4} - \frac{1}{2} \right )</math></div><br />

<div style="text-align: center;"><math>\phi \left ( r_{eq} \right ) = \epsilon</math></div><br />

=== Atomic Forces and Integrals ===
The integral below is a generalised form to which boundary conditions can be applied.

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=\int\limits_{L_1}^{L_2} 4\epsilon\left ( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6}\right )\mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r) =4\epsilon\sigma^{12} \int\limits_{L_1}^{L_2} r^{-12} \mathrm{d}r - 4\epsilon\sigma^{6}\int\limits_{L_1}^{L_2} r^{-6} \mathrm{d}r</math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{L_2} \phi(r)=4\epsilon\sigma^{6} \left [ \frac {1}{5r^5} - \frac{\sigma^{6}}{11r^{11}} \right ]_{L_1}^{L_2} </math></div><br />

The limits <math display=inline>L_1=\{2\sigma, 2.5\sigma, 3\sigma\}</math> and <math display=inline>L_2=\infty</math> can be applied as below. The final results apply for <math display=inline>\sigma=\epsilon=1</math>.

<div style="text-align: center;"><math>L_2=\infty \therefore \left ( \frac {1}{5(\infty)^5} - \frac{\sigma^{6}}{11(\infty)^{11}} \right ) = 0 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{L_1}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {\sigma^6}{11L_1^{11}} - \frac {1}{5L_1^5} \right ) = 4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11L_1^6}{55L_1^{11}} \right ) </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2\sigma)^6}{55(2\sigma)^{11}} \right ) = -\frac{699\sigma \epsilon}{28160} = -0.0248 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{2.5\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {5\sigma^6-11(2.5\sigma)^6}{55(2.5\sigma)^{11}} \right ) = -\frac{4391808\sigma \epsilon}{537109375} = -0.00818 </math></div><br />

<div style="text-align: center;"><math>\int\limits_{3\sigma}^{\infty} \phi(r)=4\epsilon\sigma^6 \left ( \frac {3\sigma^6-11(2.5\sigma)^6}{55(3\sigma)^{11}} \right ) = -\frac{32056\sigma \epsilon}{9743085} = -0.00329 </math></div><br />

=== Periodic Boundary Conditions ===
1 mL of water at STP has a mass of 1 g. Consequently, the number of moles of water in such a volume ''n'' and the number of molecules ''N'' is calculated below. The final result is ''N'' = 3.43 × 10<sup>23</sup> molecules. <br />.

<div style="text-align: center;"><math>n = \frac{m}{M_r} = \frac{1}{18.01528} = 0.0555084350618 </math></div><br />

<div style="text-align: center;"><math>N = n \cdot N_A = 0.0555084350618 \cdot 6.022140857 \times 10^{23} \approxeq 3.343 \times 10^{22} </math></div><br />

The calculation for the volume of 10000 molecules of water at STP is below. The final result is 2.99 × 10<sup>-19</sup> mL. <br />

<div style="text-align: center;"><math>n = \frac{N}{N_A} = \frac{10000}{6.022140857 \times 10^{23}} = 1.6605390404272 \times 10^-20 </math></div><br />

<div style="text-align: center;"><math>V = m = n\cdot M_r = 1.6605390404272 \times 10^{-20} \cdot 18.01528 \approxeq 2.992 \times 10^{-19} </math></div><br />

If an atom at (0.5, 0.5, 0.5) in a unit cube with repetitive boundary conditions (like the game Snake) moves along a vector of (0.7, 0.6, 0.2), it will reappear in the cube at (0.2, 0.1, 0.7).

=== Reduced Units ===
For argon, the Lennard-Jones parameters are ''σ'' = 0.34 nm and ''ϵ'' = 120''k''<sub>''B''</sub>.

<div style="text-align: center;"><math>r = r^* \cdot \sigma = 3.2 \cdot 0.34 = 1.088 \text{ nm} </math></div><br />

<div style="text-align: center;"><math>\epsilon = 120k_B = 1.656778224 \times 10^{-21} J \approxeq -2.751 \times 10^{-48} \text { kJ/mol} </math></div><br />

<div style="text-align: center;"><math>T=T^*\frac{\epsilon}{k_B} = 1.5 \cdot 120 = 180\text{ K} </math></div><br />

== Equilibration Tasks ==
=== Creating the Simulation Box ===
If atoms are generated with random coordinates, one atom could be generated in a position that overlaps with another atom. This would dramatically increase the overall energy due to intra-pair repulsive potentials, destabilising the system.

A relative lattice spacing of 1.07722 for a simple cubic lattice (1 lattice point per unit cell) generates a lattice point per unit volume density of 0.79999. Generating an atom at each lattice point for a 10×10×10 box produces 1000 atoms. A relative lattice spacing of 1.49380 for a face-centered cubic lattice (4 lattice points per unit cell) generates a lattice point per unit volume density of 1.2. Generating an atom at each lattice point for a 10×10×10 box produces 4000 atoms.

=== Setting the Properties of the Atoms ===
The following commands in LAMMPS have specific functions, as described below:

<pre>mass 1 1.0</pre> This command creates a type 1 atom with mass 1.

<pre>pair_style lj/cut 3.0</pre> This command instructs the program to compute Leonard-Jones pairwise potentials, using a cutoff point of 3.

<pre>pair_coeff * * 1.0 1.0</pre> This command gives the force field coefficients (''ϵ'' and ''σ'') between two type 1 atoms.

As the initial positions and velocities have been specified, this is consistent with the Velocity-Verlet algorithm.

=== Running the Simulation ===
Using piece of code that references the input value means that the rest of the code need not be altered when the input is altered. Furthermore, other values that depend on the given timestep can also be linked straight back to the input value.

=== Checking Equilibration ===
The simulation data of total particular energy, temperature and pressure against time for a 0.001 s timestep are graphed below (in that order). The simulation reaches a clear equilibrium for all three values, as, after a short period of time (ca. 0.03 s, or 30 timesteps).

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | Simulation Data
|-
| [[File:Mk energytime1.png|450px|center]] || [[File:Mk pressuretime1.png|450px|center]] || [[File:Mk temperaturetime1.png|450px|center]]
|}

The data for different timesteps is graphed below. It is clear that the data for the 0.001 s and 0.0025 s are very similar and can essentially be used interchangeably. In the case of simulations over long periods of time, the 0.0025 s timestep is appropriate to reduce computation time but still maintain a high level of accuracy (the mean energy value differs by 0.005%). The 0.015 s timestep does not reach an average energy value, but instead continually increases the total particular energy with time: indicative of an unstable, positively feeding-back system. The Velocity-Verlet algorithm requires the initial atomic positions and velocities, and then simulates molecular movement from there. A large timestep results in large atomic displacements per timestep, causing a significant error in the calculation which multiplies by each timestep.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of Total Particular Energies for Different Timesteps
|-
| [[File:Mk timesteptime1.png|700px|center]]
|}

== Specific Condition Simulations Tasks ==

=== Thermostats and Barostats ===
Setting γ as the velocity scaling factor to inter-convert between the instantaneous and target temperatures allows elucidation of its value in terms of the latter.

<div align="center"><math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math></div><br/>

<div align="center"><math>\frac{1}{2}\sum_i m_i \gamma^2 v_i^2 = \frac{3}{2} N k_B \mathfrak{T}</math></div><br/>

<div align="center"><math>\frac{\frac{1}{2}\sum_i m_i v_i^2}{\frac{1}{2}\sum_i m_i \gamma^2 v_i^2} = \frac{\frac{3}{2} N k_B T}{\frac{3}{2} N k_B \mathfrak{T}}</math></div><br/>

<div align="center"><math>\frac{1}{\gamma^2} = \frac{T}{\mathfrak{T}}</math></div><br/>

<div align="center"><math> \gamma = \pm \sqrt{\frac{\mathfrak{T}}{T}}</math></div><br/>

=== Examining the Input Script ===
In the program input scripts for the simulations in this section, there exists a line of code that resembles that below:

<pre> fix aves all ave/time 100 1000 100000 v_dens v_temp v_press v_dens2 v_temp2 v_press2 </pre>

The first numerical variable (100 in this case) represents the number of timesteps τ<sub>i</sub> over which an average is taken. Mathematically:

<div align="center"><math>\bar{v} _n = \frac {\sum_i \tau_i}{100}</math></div><br/>

The second numerical variable (1000 in this case) represents the number of averages over which to take a further average. Mathematically:

<div align="center"><math>\tilde{v} _i = \frac {\sum_n \bar{v} _n}{1000}</math></div><br/>

The third numerical variable (10000 in this case) represents the number of previous averages over which to take the final average. Mathematically:

<div align="center"><math>\hat{v} _a = \frac {\sum_i \tilde{v} _i}{10000}</math></div><br/>

=== Plotting the Equations of State ===

<div align="center"><math>\rho = \frac {p}{T}</math></div><br/>
The data generated from the simulations in this section have been plotted in terms of temperature and density at reduced pressures 2 and 3 in the graph below (with error bars included). The ideal density calculated by the ideal gas law in reduced units (as above) is also plotted. The graph shows a clear discrepancy between the predicted and simulated densities, which is due to the ideal gas law treating the atoms as classical hard balls that experience no repulsion unless they are in direct contact (bouncing off each other). However, the Lennard-Jones potential models the atoms more faithfully to reality, in that, within a critical radius, the electrons orbiting the atoms repel each other and the atoms experience net repulsion. The densities predicted by the ideal gas are consequently higher as they do not take neighbour repulsions into consideration, which means that the model predicts that the atoms can be within closer proximities of one another without destabilising the system.

However, it is also clear that the discrepancy decreases with increasing temperature. This is because, at higher temperatures, the atoms have more kinetic energy and thus behave more like Newtonian hard balls, as their kinetic energy is dominant over the repulsive electronic forces until a smaller distance. Essentially, the atoms can get closer as their kinetic energy can overcome the electronic repulsion.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Computed and Ideal Densities at varying Temperatures and Pressures
|-
| [[File:Mk temperaturedensity1.png|700px|center]]
|}

== Statistical Physics Tasks ==

In this section, ten simulations were again run to determine the change in heat capacity per unit volume for a liquid at varying densities. An exemplar script can be found [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Script_examples_with_milandeleev here]. The results are graphed below. Higher density very obviously produces a higher heat capacity, which is expected because more atoms in one space can absorb more heat energy than fewer atoms in the same space. There is a clear decrease in the heat capacity with increasing temperature, which is at odds with the physical predictions. At lower temperatures, there are fewer thermally accessible translational, rotational, electronic and vibrational energy levels available. As temperature increases, more of these states become available so more of the energy levels become populated, making the internal energy rise rapidly. Since the heat capacity is the derivative of the internal energy with respect to temperature, higher temperatures are thus expected to increase the heat capacity (although this does level out towards the phase transition temperature). This deviation from the expected trend could possibly be explained by the fact that this system is modelling simple hydrogen atom fluids, and thus rotational and vibrational energy levels are not available. Furthermore the software itself may not take into account electronic transitions.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="1" | Comparison of the Variation of Heat Capacities per Unit Volume with Temperature for Varying Densities
|-
| [[File:Mk temperatureheatcap1.png|700px|center]]
|}

== Radial Distribution Function Tasks ==

The radial distribution functions of multiple phases and their integrals are plotted against distance below. The radial distribution function for these simulations should average out to one, irrespective of the phase, because the function itself is a measure of the particle density surrounding a given particle. Consequently it can be seen as a measure of how consistently structured a material is: for a gas, the position of the particles is completely unpredictable and so it will very quickly tend to a central value. However, for a crystalline solid, the peaks will average unity but will never become a straight line that tend to unity, because the crystal has a defined structure with particles a fixed distance away from each other. Henceforth, there is a much greater probability that a particle will be found in a lattice site than outside of one, so the RDF will never lose its 'bumpiness'. From this line of reasoning, the RDF of a simple liquid should tend to unity but slower than the same function for a gas.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | RDF and Integrated RDF for Three Phases
|-
| [[File:Mk rdfr1.png|450px|center]] || [[File:Mk intrdfr1.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

To find the coordination number, the value of the first minimum on the RDF of the solid is found (which appears at <math display="inline">r=1.975</math>), which equates to a coordination number of 12. The lattice spacing, consequently, is ca. 1.37.

== Dynamical Properties Tasks ==

The mean squared displacements for a small sample and a large sample in different phases are plotted below.

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="2" | Mean Squared Displacement for Varying Phases and Sample Sizes
|-
| Fewer Atoms || One Million Atoms
|-
| [[File:Mk msdt1.png|450px|center]] || [[File:Mk msdt2.png|450px|center]]
|-
| colspan="2" | [[File:Mk key1.png|200px]]
|}

The resulting diffusion coefficients, calculated by the equation below (in which ''n'' is the dimensionality, or 3 in this case), are also tabulated below.

<div align="center"><math> \frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}=2nD</math></div>

{| class="wikitable floatcentre" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! colspan="3" | <math>D</math> (m<sup>2</sup>s<sup>-1</sup>) to 8 s.f
|-
| Phase || Small Sample || Large Sample
|-
| Solid || 0.010172398 || 5.4978008 × 10<sup>-7</sup>
|-
| Liquid || 0.19168932 || 0.088731507
|-
| Gas || 2.7676196 || 3.0690123
|}

In general, these findings are logically consistent. The mean-squared displacement and thus the diffusion coefficient should be highest for a gas, as the particles have a higher kinetic energy and so move away from their original positions fastest. Liquids have lower diffusion coefficients, and solids lower still. However, comparing the simulation for fewer and more atoms reveals some odd data. The diffusion coefficients for the gas phase are relatively similar, with a small increase on the side of the larger sample. This difference in value can simply be attributed to the greater accuracy of the million-atoms calculation. However, the value of <math display=inline>D</math> for the liquid simulation is smaller for one million atoms, and for a solid it is much smaller still (by a magnitude of one hundred thousand). The values should at least be similar, as the parameters are similar, but the sample sizes are different. To rationalise this, it is possible to consider that the value for <math display=inline>D</math> for one million atoms as a solid is skewed due to the presence of negative micro-gradient values, so it is difficult to compare in any valid sense to the computation for fewer atoms. For the case of the liquid, the origin of the difference is difficult to reason.

Rep:Mod:cej15 Transition States

2018-02-28T03:31:58Z

Cej15: /* Energies and Analysis */

==Introduction==
===Transition State===
A transition state of a particular reaction is the point where the reaction reaches its maximum potential energy. The activated complex would then either proceed to form the product, or return to its reagents.
===Potential Energy Surface===
The transition state can also be defined as a surface in configuration space that divides reactants from products and passes through the saddle point of the potential-energy surface. [ref: Donald G. Truhlar, and Bruce C. Garrett Acc. Chem. Res., 1980, 13 (12), pp 440–448]

==Exercise 1: Reaction of Butadiene with Ethylene==
===Optimisation and Determination of Transition State===
Both structures for the reactants and the structure for the product were constructed in Gaussian and optimised to a PM6 level. The bond lengths and bond angles for the optimised product were altered to resemble the transition state, and this assumed transition state was then optimised to a Berny Transition State.
In order to confirm if the transition state was correct or not, frequency calculations were made and the Intrinsic Reaction Coordinate was determined. The frequency of the transition state was -949.59cm-1 and the gifs for the vibrations and IRC are shown below.

===MO Analysis===
For two molecular orbitals to interact, they must be of the same symmetry, which could also be shown in the table below, meaning that antisymmetric orbitals can only interact with antisymmetric orbitals and same for symmetric ones. This is because MOs would interact to form a non-zero overlap orbital. Mathematically, an antisymmetric function would have an overall integral of zero and combining a symmetric function and an antisymmetric function would produce this result, and this is why this kind of interacting is forbidden. Therefore, symmetric-antisymmetric interactions produce zero overlapping while symmetric-symmetric and antisymmetric-antisymmetric interactions produce non-zero overlaps.
{| class="wikitable" style="text-align: center" border=1
! Ethene MO !! 1,3-Butadiene MO || Symmetry || Transition State MOs || MO Diagram
|-
| rowspan="2" | [[File:ETHYLENE_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:BUTADIENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>a</math> || [[File:HIGHEST_MO_CEJ1.jpg|thumb|center|<div style="text-align: center">Highest MO]] || rowspan="4" | [[File:MO DA MK1.png|500px]]
|-
| [[File:TS_lowest_MO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO+1]]
|-
| rowspan="2" | [[File:ETHYLENE_HOMO_cej.jpg|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:BUTADIENE_LUMO_cej.jpg|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>s</math> || [[File:TS_HOMO_CEJ1.jpg|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:TS_LUMO_cej1.jpg|thumb|center|<div style="text-align: center">LUMO]]
|}
===Bond Length Analysis===

Shown below is a table of the bond lengths for each substance related in this reaction. A typical sp<sub>2</sub> C-C bond length is around 147 pm and a carbon double bond is around 133 pm, as shown for the bond lengths for butadiene and ethylene. A typical sp<sub>3</sub> C-C bond length is around 154 pm. Shown below is also a graph showing the change in bond lengths during the whole reaction.

{| style="text-align: center; " border=1
|+ '''Bond Lengths (Unit = pm)'''
! Carbons !! Butadiene !! Ethylene !! Transition State !! Product
|-
| C1-C2 || 134 || 133 || 138 || 149
|-
| C2-C3 || 147 || || 211 || 154
|-
| C3-C4 || 134 || || 138 || 154
|-
| C4-C5 || || || 211 || 154
|-
| C5-C6 || || || 138 || 149
|-
| C6-C1 || || || 141 || 133
|}

[[File:C_bond_change.png|600px]]

===Log File Uploads===
[[:File:BUTADIENE_OPT.LOG|Butadiene]]

[[:File:Ethene_opt_cej.log|Ethylene]]

[[:File:TS_opt_cej1.log|Transition State]]

[[:File:IRC_cej1.log|Intrinsic Reaction Coordinate]]

[[:File:PRODUCT_OPT_CEJ1.LOG|Cyclohexene]]

==Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole==
===Optimisation and Determination of Transition State===
===MO Analysis===

{| class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto" border=1
! Cyclohexadiene MO !! 1,3-Dioxole MO || Symmetry || Endo Transition State MOs || Exo Transition State MOs
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | [[File:DIOXOLE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | <math>a</math> || [[File:ENDO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]] || [[File:EXO_LOWEST_CEJ.PNG|thumb|center|<div style="text-align: center">LOWER MO]]
|-
| [[File:ENDO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]] || [[File:EXO_HIGHEST_CEJ.PNG|thumb|center|<div style="text-align: center">HIGHER MO]]
|-
| rowspan="2" | [[File:CYCLOHEXADIENE_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || rowspan="2" | [[File:DIOXOLE_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || rowspan="2" | <math>s</math> || [[File:ENDO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]] || [[File:EXO_HOMO_CEJ.PNG|thumb|center|<div style="text-align: center">HOMO]]
|-
| [[File:ENDO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]] || [[File:EXO_LUMO_CEJ.PNG|thumb|center|<div style="text-align: center">LUMO]]
|-
|}

===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Cyclohexadiene || -233.32
|-
| 1,3-Dioxole || -267.07
|-
| rowspan ="2" | Product || Exo Diels-Alder Adduct || -500.43
|-
| Endo Diels-Alder Adduct || -500.42
|}
The table to the right shows the sum of electronic and thermal free energies of the reactants and the products. This can be used to calculate the change in the standard Gibbs Free Energy for both reactions using the equation below:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = \sum_{products} (\varepsilon_0 + G_{corr}) - \sum_{reactants} (\varepsilon_0 + G_{corr})</math></div>

which turns out to be:

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.43 - (-233.32 + -267.07) = -0.04 kJ/mol</math></div> for the exo adduct and,

<div align="center"><math>\Delta_rG^{\ominus}(298.15K) = -500.42 - (-233.32 + -267.07) = -0.03 kJ/mol</math></div> for the endo adduct.

The two is very similar as shown, with a very small difference of 0.01 kJ/mol. This suggests that both is favourable, but due to the fact that the exo adduct did have a larger change in gibbs free energy, the exo adduct is supposed to be more thermodynamically stable, while the endo adduct is more kinetically stable. This could be due to the endo adduct having better orbital overlaps than the exo adduct, but the exo adduct reduces repulsion more than the endo adduct.

===Log File Uploads===
[[:File:CYCLOHEXADIENE_OPT_CEJ.LOG|Cyclohexadiene]]

[[:File:DIOXOLE_OPT_CEJ.LOG|Dioxole]]

[[:File:ENDO_TS_CEJ.log|Transition State for Endo Product]]

[[:File:EXO_TS_CEJ.log|Transition State for Exo Product]]

[[:File:ENDO_IRC_CEJ.log|Intrinsic Reaction Coordinate of Endo Product]]

[[:File:EXO_IRC_cej.log|Intrinsic Reaction Coordinate of Exo Product]]

[[:File:ENDO_product_cej.log|Endo Product]]

[[:File:EXO_PRODUCT_CEJ.LOG|Exo Product]]

==Exercise 3: Diels-Alder vs Cheletropic==
===Optimisation and Determination of Transition State===
===Energies and Analysis===

{| class="wikitable floatright" style="text-align: center" border=1
|+ '''Reactant and Product Energies (kJmol<sup>-1</sup>)'''
! !! Compound || <math>\varepsilon_0 + G_{corr}</math>
|-
| rowspan="2" | Reactant || Xylylene || -309.50
|-
| Sulfur Dioxide || -548.60
|-
| rowspan ="2" | Product || Diels-Alder || -fesfesfefesfesfse
|-
| Cheletropic || -853.53
|}
The energy values of the reactants and products are shown in the table on the right.

===Log File Uploads===
[[:File:SO2_opt_cej.log|Sulfur Dioxide]]

[[:File:XYLYLENE_OPT_CEJ.LOG|Xylylene]]

[[:File:DA_TS_CEJ3.log|Diels-Alder Transition State]]

[[:File:Chele_TS_cej.log|Cheletropic Transition State]]

[[:File:DA_IRC_cej3.log|Diels-Alder Intrinsic Reaction Coordinate]]

[[:File:Chele_IRC_cej.log|Cheletropic Intrinsic Reaction Coordinate]]

[[:File:DA_PRODUCT_cej3.log|Diels-Alder Product]]

[[:File:PRODUCT_CHELETROPIC_CEJ.LOG|Cheletropic Product]]

Rep:Mod:cej15 Transition States

2018-02-28T03:28:08Z

Cej15: /* Exercise 3: Diels-Alder vs Cheletropic */

Rep:Mod:cej15 Transition States

2018-02-28T03:23:11Z

Cej15: /* Energies and Analysis */

Rep:Mod:cej15 Transition States

2018-02-28T03:00:02Z

Cej15: /* Exercise 2: Reactions of Cyclohexadiene and 1,3-Dioxole */

Rep:Mod:cej15 Transition States

2018-02-28T02:53:57Z

Cej15: /* MO Analysis */

Rep:Mod:cej15 Transition States

2018-02-28T02:48:59Z

Cej15: /* MO Analysis */

Rep:Mod:cej15 Transition States

2018-02-28T02:46:55Z

Cej15: /* MO Analysis */

File:EXO LUMO CEJ.PNG

2018-02-28T02:41:29Z

Cej15:

File:EXO LOWEST CEJ.PNG

2018-02-28T02:41:03Z

Cej15:

File:EXO HOMO CEJ.PNG

2018-02-28T02:40:43Z

Cej15:

File:EXO HIGHEST CEJ.PNG

2018-02-28T02:40:28Z

Cej15:

File:ENDO LUMO CEJ.PNG

2018-02-28T02:40:13Z

Cej15:

File:ENDO LOWEST CEJ.PNG

2018-02-28T02:39:59Z

Cej15:

File:ENDO HOMO CEJ.PNG

2018-02-28T02:39:44Z

Cej15:

File:ENDO HIGHEST CEJ.PNG

2018-02-28T02:39:25Z

Cej15:

File:DIOXOLE LUMO CEJ.PNG

2018-02-28T02:37:11Z

Cej15: