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MRD:mc4717

2019-05-16T17:02:23Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions.
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected.

===Polanyi's Empirical Rules<ref name="TS_theory" />===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T17:02:04Z

Mc4717: /* Transition State Theory */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions.
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules<ref name="TS_theory" />===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T17:00:43Z

Mc4717: /* Transition State Theory */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules<ref name="TS_theory" />===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T17:00:02Z

Mc4717: /* Polanyi's Empirical Rules */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules<ref name="TS_theory" />===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:59:32Z

Mc4717: /* Transition State Theory */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:59:09Z

Mc4717: /* Transition State Theory */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory <ref name="TS_Theory" />===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:58:33Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
 As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:58:08Z

Mc4717: /* F + H2 → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.<ref name="hydrogen_bond" />

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:55:33Z

Mc4717: /* Internuclear Distance against Time Plot */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.<ref name="hydrogen_bond" />

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:54:57Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

== References ==
<references>
<ref name="hydrogen_bond"> P. Atkins and J. De Paula, ''Atkins' Physical chemistry'', Oxford University Press, Oxford, 2006.</ref>
<ref name="TS_theory">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
</references>

MRD:mc4717

2019-05-16T16:49:33Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

<references>
polanyi
transition state
hydrogen biond
bond energies
P. Atkins and J. De Paula, Atkins' Physical chemistry, Oxford University Press, Oxford, 2006.
</references>

MRD:mc4717

2019-05-16T16:42:25Z

Mc4717: /* Dynamics from the transition state region */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:32:26Z

Mc4717: /* Transition State Theory */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative is shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:
 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.

==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:32:15Z

Mc4717: /* Reactive and unreactive trajectories */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative is shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HBHC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:31:40Z

Mc4717: /* Reactive and unreactive trajectories */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative is shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:31:16Z

Mc4717: /* Reactive and unreactive trajectories */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative is shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HAHB does not oscillate and after the reaction HBHC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HBHC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HAHB collide and the HAHB bond lengthens however, as the vibrational energy is too high HAHB reforms and HBHC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HAHB + HC → HBHC + HA. HA is ejected and HBHC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:29:52Z

Mc4717: /* Dynamics from the transition state region */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative is shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:27:35Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred.
>br>As above the energy deposited in the system is vibrational energy, efficient for this system and sufficient to drive the reaction.

To conclude, increasing the translational energy promotes an exothermic reaction and increasing the vibrational energy promotes an endothermic reaction; the tests carried out are in accordance with the Polyani's rules.

MRD:mc4717

2019-05-16T16:24:30Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

As seen in Figure 18 and the system conditions, by increasing the amount of vibrational energy in the system (pFH) resulted in a successful collision and product formation. As the reactants have sufficient KE the activation energy barrier was crossed and the chemical reaction occurred. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

MRD:mc4717

2019-05-16T16:21:27Z

Mc4717: /* Reverse Reaction: HF + H → H2 + F */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

As shown on Figure 17 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is insufficient KE to overcome the Ea barrier and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is effective for this system, as the reaction is endothermic with a late-barrier reaction and a TS energetically closer to the products. In this case the vibrational energy is insufficient to drive the reaction.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

MRD:mc4717

2019-05-16T16:10:30Z

Mc4717: /* Reverse Reaction: HF + H → H2 + F */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

MRD:mc4717

2019-05-16T16:07:39Z

Mc4717: /* References */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

MRD:mc4717

2019-05-16T16:05:22Z

Mc4717: /* Promoting the backward reaction: HA + HB-F → HA-HB + F */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T16:04:43Z

Mc4717: /* Test 2 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 16 H2 and F collide and the reaction does go to completion as the collision is successful. Although the total energy of the system is lowered (lower vibrational energy) the translation energy of the reactants is raised as seen in the reaction conditions and the products are formed.
 The energy deposited in the system is translational energy; as per Polyani's Rules translational energy is efficient for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is promoted.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:55:57Z

Mc4717: /* Test 1 */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.
 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

This is because most of the energy is deposited in the reactants as H-H vibrations as seen by the large oscillations in H2. Barrier recrossing has occured as a result of the large vibrational energy causing the system to revert back to the reactants even though the system has already crossed the transition state and a bond was formed between HB and F temporarily.

2. The initial system conditions used are: rHH = 0.74 Å, rHF = 1.8 Å, pHH = 0.1 and pHF = -0.8.

[[File:01409469 polyani 3+4 2.jpg|thumb|center|800px|'''Figure 19.''' Contour plot of the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 '''Figure 20.''' Graph of momentum against time for the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 (Left-right)]]

The collision between a molecule of H2 and F is successful and a chemical reaction has occurred. A reaction has occurred even though the overall energy of the system has been significantly reduced by lowering the amount of vibrational energy in the reactants but slightly increasing the amount of translational energy deposited in the reactants. Therefore, H2 and F have more translational energy to move towards each other and form a H-F bond. The product formed is vibrationally excited as seen by the large amplitude of oscillations in the product HF which correspond to the strong H-F vibrations.

From the second simulation, it can be seen that an increase in translational energy in the reactants is more effective at promoting an exothermic reaction with an early transition state to occur, which again follows Polyani's rules.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:55:03Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy; as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants. Therefore the forward exothermic reaction is not promoted.

As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

This is because most of the energy is deposited in the reactants as H-H vibrations as seen by the large oscillations in H2. Barrier recrossing has occured as a result of the large vibrational energy causing the system to revert back to the reactants even though the system has already crossed the transition state and a bond was formed between HB and F temporarily.

2. The initial system conditions used are: rHH = 0.74 Å, rHF = 1.8 Å, pHH = 0.1 and pHF = -0.8.

[[File:01409469 polyani 3+4 2.jpg|thumb|center|800px|'''Figure 19.''' Contour plot of the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 '''Figure 20.''' Graph of momentum against time for the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 (Left-right)]]

The collision between a molecule of H2 and F is successful and a chemical reaction has occurred. A reaction has occurred even though the overall energy of the system has been significantly reduced by lowering the amount of vibrational energy in the reactants but slightly increasing the amount of translational energy deposited in the reactants. Therefore, H2 and F have more translational energy to move towards each other and form a H-F bond. The product formed is vibrationally excited as seen by the large amplitude of oscillations in the product HF which correspond to the strong H-F vibrations.

From the second simulation, it can be seen that an increase in translational energy in the reactants is more effective at promoting an exothermic reaction with an early transition state to occur, which again follows Polyani's rules.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:52:15Z

Mc4717: /* Polanyi's Empirical Rules */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

<li>Vibrational energy→pHH
<li>Translational energy→pHF

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy therefore, as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants.

As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.
This appears to run counter to what one would expect to happen when more energy is put into the system. It is predicted that when more energy is put into the system, the reactants would possess more kinetic energy to overcome the activation energy barrier. However, as can be seen from the simulation above, an increase in the vibrational energy of the reactants is not effective at promoting the occurrence of an exothermic reaction with an early transition state. This is in accordance to Polyani's rules.

This is because most of the energy is deposited in the reactants as H-H vibrations as seen by the large oscillations in H2. Barrier recrossing has occured as a result of the large vibrational energy causing the system to revert back to the reactants even though the system has already crossed the transition state and a bond was formed between HB and F temporarily.

2. The initial system conditions used are: rHH = 0.74 Å, rHF = 1.8 Å, pHH = 0.1 and pHF = -0.8.

[[File:01409469 polyani 3+4 2.jpg|thumb|center|800px|'''Figure 19.''' Contour plot of the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 '''Figure 20.''' Graph of momentum against time for the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 (Left-right)]]

The collision between a molecule of H2 and F is successful and a chemical reaction has occurred. A reaction has occurred even though the overall energy of the system has been significantly reduced by lowering the amount of vibrational energy in the reactants but slightly increasing the amount of translational energy deposited in the reactants. Therefore, H2 and F have more translational energy to move towards each other and form a H-F bond. The product formed is vibrationally excited as seen by the large amplitude of oscillations in the product HF which correspond to the strong H-F vibrations.

From the second simulation, it can be seen that an increase in translational energy in the reactants is more effective at promoting an exothermic reaction with an early transition state to occur, which again follows Polyani's rules.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:51:58Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]

 As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy therefore, as per Polyani's Rules vibrational energy is not effective for this system, as the reaction is exothermic with an early-barrier reaction and a TS energetically closer to the reactants.

As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.
This appears to run counter to what one would expect to happen when more energy is put into the system. It is predicted that when more energy is put into the system, the reactants would possess more kinetic energy to overcome the activation energy barrier. However, as can be seen from the simulation above, an increase in the vibrational energy of the reactants is not effective at promoting the occurrence of an exothermic reaction with an early transition state. This is in accordance to Polyani's rules.

This is because most of the energy is deposited in the reactants as H-H vibrations as seen by the large oscillations in H2. Barrier recrossing has occured as a result of the large vibrational energy causing the system to revert back to the reactants even though the system has already crossed the transition state and a bond was formed between HB and F temporarily.

2. The initial system conditions used are: rHH = 0.74 Å, rHF = 1.8 Å, pHH = 0.1 and pHF = -0.8.

[[File:01409469 polyani 3+4 2.jpg|thumb|center|800px|'''Figure 19.''' Contour plot of the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 '''Figure 20.''' Graph of momentum against time for the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 (Left-right)]]

The collision between a molecule of H2 and F is successful and a chemical reaction has occurred. A reaction has occurred even though the overall energy of the system has been significantly reduced by lowering the amount of vibrational energy in the reactants but slightly increasing the amount of translational energy deposited in the reactants. Therefore, H2 and F have more translational energy to move towards each other and form a H-F bond. The product formed is vibrationally excited as seen by the large amplitude of oscillations in the product HF which correspond to the strong H-F vibrations.

From the second simulation, it can be seen that an increase in translational energy in the reactants is more effective at promoting an exothermic reaction with an early transition state to occur, which again follows Polyani's rules.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:47:42Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]
Reaction goes to completion

Vibrational energy is represented by pHH while translational energy is represented by pHF.

As shown on Figure 15 H2 and F collide however, the reaction does not go to completion as the collision is unsuccessful. Although the molecules have collided and there is sufficient KE to overcome the Ea barrier, barrier recrossing occurs and the products are not formed.

 The energy deposited in the system is vibrational energy therefore, as per Polyani's Rules. This appears to run counter to what one would expect to happen when more energy is put into the system. It is predicted that when more energy is put into the system, the reactants would possess more kinetic energy to overcome the activation energy barrier. However, as can be seen from the simulation above, an increase in the vibrational energy of the reactants is not effective at promoting the occurrence of an exothermic reaction with an early transition state. This is in accordance to Polyani's rules.

This is because most of the energy is deposited in the reactants as H-H vibrations as seen by the large oscillations in H2. Barrier recrossing has occured as a result of the large vibrational energy causing the system to revert back to the reactants even though the system has already crossed the transition state and a bond was formed between HB and F temporarily.

2. The initial system conditions used are: rHH = 0.74 Å, rHF = 1.8 Å, pHH = 0.1 and pHF = -0.8.

[[File:01409469 polyani 3+4 2.jpg|thumb|center|800px|'''Figure 19.''' Contour plot of the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 '''Figure 20.''' Graph of momentum against time for the reaction between F and H2 when pHH = 0.1 and pHF = -0.8 (Left-right)]]

The collision between a molecule of H2 and F is successful and a chemical reaction has occurred. A reaction has occurred even though the overall energy of the system has been significantly reduced by lowering the amount of vibrational energy in the reactants but slightly increasing the amount of translational energy deposited in the reactants. Therefore, H2 and F have more translational energy to move towards each other and form a H-F bond. The product formed is vibrationally excited as seen by the large amplitude of oscillations in the product HF which correspond to the strong H-F vibrations.

From the second simulation, it can be seen that an increase in translational energy in the reactants is more effective at promoting an exothermic reaction with an early transition state to occur, which again follows Polyani's rules.

=== Promoting the backward reaction: HA + HB-F → HA-HB + F ===

1. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 0.2 and pHH = -3.0.

[[File:01409469 polyani 5+6 .jpg|thumb|center|800px|'''Figure 21.''' Contour plot of the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 '''Figure 22.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 0.2 and pHH = -3.0 (Left-right)]]

An atom of HA collides with a molecule of HBF. However, the collision is unsuccessful and no chemical reaction occurs. This is because the reactants lack sufficient vibrational energy to surmount the activation energy barrier. Therefore, following an unsuccessful collision with HBF, HA simply bounces off and moves away at constant momentum. This is in agreement with Polyani's rules which state that vibrational energy is more effective than translational energy at promoting reactions with a late transition state, for example, endothermic chemical reactions.

2. The initial system conditions used are: rFH = 0.92 Å, rHH = 2.0 Å, pFH = 7.1 and pHH = -0.5.

[[File:01409469 polyani 7+8 .jpg|thumb|center|800px|'''Figure 23.''' Contour plot of the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 '''Figure 24.''' Graph of momentum against time for the reaction between HA + HB-F when pFH = 7.1 and pHH = -0.5 (Left-right)]]

Increasing the amount of vibrational energy in the reactants by increasing pFH resulted in a chemical reaction successfully occurring. This is because the reactants now have sufficient kinetic energy in the form of H-F vibrations to overcome the activation energy barrier. In addition, because the reaction between HA + HB-F is endothermic, energy is taken in from the surroundings during the reaction and the products are less vibrationally excited than the reactants. This can be seen in the lower amplitude of oscillations in the product formed.

In conclusion, the vibrational energy of the reactants should be increased to promote an endothermic reaction with a late transition state while the translational energy of the reactants should be increased to promote an exothermic reaction with an early transition state. All 4 simulations shown above obey Polyani's rules.

== References ==
<references>
<ref name="ref1">A. C. Vaucher, M. Reiher, ''J. Chem. Theory. Comput.'', 2018, '''14''', 3091-3099.</ref>
<ref name="ref2">J. I. Steinfield, J. S. Francisco, W. L. Hase, ''Chemical Kinetics and Dynamics'', Pearson, 1998.</ref>
<ref name="ref3">D. C. Elton, ''Transition State Theory for Physicists'', 2013.</ref>
<ref name="ref4">Z. Zhang, Y. Zhou, D. H. Zhang, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:32:45Z

Mc4717: /* Reverse Reaction: HF + H → H2 + F */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]
Reaction goes to completion

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|400px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|400px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:32:31Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|400px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|400px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]
Reaction goes to completion

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|800px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|800px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:32:09Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|600px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|600px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]
Reaction goes to completion

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|800px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|800px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:31:17Z

Mc4717:

==Molecular Reaction Dynamics Computational Lab==
==H + H2 system==
===Dynamics from the transition state region===
The transition state is defined as "the maximum on the minimum energy path linking reactants and products."
Therefore, the transition state can be found as a stationary point on the potential energy surface graph and the derivative of the potential energy is zero with respect to the Cartesian coordinates.
<math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math>
However, on the potential energy surface a local minimum point has the same value as <math>\tfrac{\partial V(r_i)}{\partial r_i}\ = 0 </math> therefore, the second derivative <math>\tfrac{\partial^2 V(r_i)}{\partial r_i ^2}\ </math> is calculated to differentiate the transition state from local minima.
The Hessian matrix used to calculate the determinant and the second derivative are shown below:
:<math>\mathbf H =
\begin{bmatrix}
\dfrac{\partial^2 \ V(r_i)}{\partial \ x^2} & \dfrac{\partial^2 \ V(r_i)}{\partial \ x \partial \mathbf y} \\
\dfrac{\partial^2 \ V(r_i)}{\partial \ y \partial \mathbf x} & \dfrac{\partial^2 \ V(r_i)}{\partial \mathbf y^2}
\end{bmatrix}</math>
Mathematically, the transition state is a saddle point and according to the results of the Hessian Matrix the transition state can be identified as follows:
<li>(x,y) is a local maximum when, <math>f</math> < 0 (1 negative eigenvalue).
<li>(x,y) is a local minimum when, <math>f</math> > 0 (1 positive eigenvalue).
<li>(x,y) is a saddle point when, <math>f</math> has both negative and positive eigenvalues.

=== Trajectories from r1 = r2: locating the transition state (rts)===

[[File:Mc_transition_contourplot.PNG|thumb|center|500px|'''Fig. 1''' Contour plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]
[[File: mc_transition_surfaceplot.PNG|thumb|center|500px|'''Fig. 2''' Surface plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

After observing the potential energy surface, the transition state was estimated to be approximately at 0.9 Å. The values for the energy against time plot were exported, in order to evaluate which value lies between the minima and the maxima. The transition point was therefore, estimated to lie between the values 0.9-9.1 Å. In order to acquire values corresponding to the transition state (KE = 0) the distances were made more accurate by increasing the decimal points of 9.0 Å until the value rts=0.9077424 was acquired.

[[File: mc_transition_distance_timeplot.PNG|thumb|center|800px|'''Fig. 3'''Internuclear distance against time plot when rBC=rAB=0.9077424 Å and pBC=pAB=0]]

At rts=0.9077424 Å the line in the internuclear distance against time plot is horizontal which, means that the molecules are at the transition state and vibrating symmetrically. Furthermore, when rBC=rAC=0.9077424 Å and pBC=pAC=0, the forces along AB and BC and the kinetic energy are 0. As there is no initial momentum on the transition state the values confirm that HA and HBC are stationary and the transition state is confirmed.

=== Calculating the reaction path: MEP and Dynamics ===
After having located the transition state position the minimum energy reaction (MEP) path trajectory can be calculated.
 The minimum energy path is the path form reactants to products with the lowest energy and it was determined by using the initial conditions to find the transition state position, except that the internuclear distance of HA and HB was increased to 0.9177424 Å. The reaction path was obtained using the calculation types Dynamics and MEP and analysed.
====Surface Plot====
[[File: mc_mep_surfaceplot.PNG|thumb|center|400px|'''Fig. 4''' Surface plot of the trajectory using MEP]]
[[File: mc_dynamic_surfaceplot.PNG|thumb|center|400px|'''Fig. 5''' Surface plot of the trajectory using dynamics]]
The reaction path can be detected on the surface plot as the black line. Based on the MEP calculation method, the trajectory as shown on the graph follows a relatively straight line, indicating that the atoms are not vibrating (KE=0). As the momenta are set to zero after each step in the minimum energy path, the reactants are converted into products infinitely slowly.
However, based on the dynamic calculation method, the trajectory as shown on the graph follows a wavy line, indicating that the atoms are oscillating (KE≠0). This is because at regions of high potential the reactants convert to products.
 As the mass and inertia of the atoms are neglected in the MEP calculation model, it does not represent a realistic model of the atom motion in the reaction progress.
====Internuclear Distance against Time Plot====
[[File: mc_mep_distance_timeplot.PNG|thumb|center|500px|'''Fig. 6''' Internuclear distance against steps using MEP]]
[[File: mc_dynamic_distance_timeplot.PNG|thumb|center|500px|'''Fig. 7''' Internuclear distance against steps using dynamics]]

In the MEP graph, the distance between HA and HB decreases during the first 50 steps and it plateaus to a value of ca. 0.75 Å (hydrogen bond length). This means signifies a bond formation HA-HB. The distance between HB and HC increases at large step values which signifies that that HC is removed after HA and HBC collide successfully.
 In the dynamic graph, the distance between HA and HB decreases, which signifies the hydrogen bond formation between HA and HB, whereas the distance between HB and HC increases.
 A difference between the figures is the internuclear distance between HA and HB. Whereas in the MEP the distance is constant because the bond is static in the dynamic the internuclear distance is oscillating, because the bond is vibrating.

=== Reactive and unreactive trajectories ===

The initial conditions were r1 = 0.74 Å and r2 = 2.0 Å, with p'''1''' and p'''2''' varied as shown in the table below.
{| class="wikitable" border=1
! p1 !! p2 !! Etot !! Contour plot !! Reactivity !! Dynamics description
|-
| -1.25 || -2.5 ||-99.018 ||[[File:mc_table1.PNG|thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. Initially HA-HB does not oscillate and after the reaction HB-HC oscillates and HA is ejected.
|-
| -1.5 || -2.0 ||-100.456 ||[[File: mc_table2.PNG |thumb|center|400px]] ||Unreactive ||The KE is insufficient and the molecule cannot overcome the activation energy barrier therefore, the collision is unsuccessful and a HB-HC bond does not form.
|-
| -1.5 || -2.5 ||-98.956 ||[[File: mc_table3.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HB-HC oscillates and HA is ejected. (KE overcomea Ea.
|-
| -2.5 || -5.0 ||-84.956 ||[[File: mc_table4.PNG |thumb|center|400px]] ||Unreactive ||
HC + HA-HB collide and the HA-HB bond lengthens however, as the vibrational energy is too high HA-HB reforms and HB-HC does not form.
|-
| -2.5 || -5.2 ||-83.416 ||[[File: mc_table5.PNG |thumb|center|400px]] ||Reactive || HA-HB + HC → HB-HC + HA. HA is ejected and HB-HC due to high, initial KE has high vibrational energy.
|}

As demonstrated in the above table, a successful reaction does not depend solely on the KE being larger than the Ea. In example 4, although the KE was sufficient the reaction was unsuccessful, whereas in examples 1 and 3 a reaction occurred despite the lower energy systems.

=== Transition State Theory ===

Transition state theory is a theory that provides the reaction rate and rate constant in elementary chemical reactions. This model involves a chemical equilibrium (quasi-equilibrium) between the reactants and the activated transition state complex and the TS, reactants and product interactions. <ref name="ref2" />:

 1. The Maxwell-Boltzmann distribution governs the distribution of reactant molecules in their respective states.
 2. As the Born-Oppenheimer approximation postulates electronic and nuclear motions are separated.
 3. The activated transition state complex and the reactants are in equilibrium.
 4. Once reactants cross the transition state the products form and reactants cannot be reformed.
 5. Particles are treated classically (quantum phenomena are not expected).

The assumptions stated do not conform with reality and limitations arise in the transition state theory.ref name="ref3" />
 Firstly, assumption 4 states that once the reactants cross the activation energy barrier the products will form. However, as seen in example 4 in the table above the system reformed the reactants although the KE was sufficient to form the products and the transition state was crossed; this is called barrier recrossing. Therefore, the rate values expected are greater than the obtained experimental ones.
 Secondly, assumption 5 neglects the implication of quantum phenomena. For lighter atom and lower activation energy systems quantum effects are considerable. When the activation energy barrier of a system is low and the KE of the reactant is insufficient to cross the TS barrier tunnelling can occur leading to product formation. The transition state theory neglects such phenomena and the rate values expected are lower than the obtained experimental ones.
==F-H-H system==

=== PES inspection===

====F + H2 → HF + H====

[[File: mc_surface_endo.PNG |thumb|center|400px|'''Fig. 8''' PES plot for the reaction F + H2]].
The reaction between F and H2 is exothermic. The reaction involves the breaking of a H2 bond and the formation of a HF bond. The H-H bond is 436 kJ/mol [6], hence weaker than the H-F bond of 565 kJ/mol. This is due to the higher EN of the F which results in a higher polar covalent bond forming. As the bond that is formed is stronger than the bond being broken the reaction is overall exothermic and the reactants are higher energetically than the products. As seen IN Fig. 8 the reactants have a greater potential energy than the products.

====H + H-F → H2 + F====

[[File:mc_surface_exo.PNG|thumb|center|400px|'''Fig. 9''' PES for the reaction H + HF]]
The reaction between HF and H is endothermic. The reaction involves the breaking of a HF bond and the formation of a H2 bond. The H-F bond is 565 kJ/mol [6], hence stronger than the H-H bond of 436 kJ/mol. This is due to the homonuclear bond formed which is not polar covalent. As the bond that is formed is weaker than the bond being broken the reaction is overall endothermic and the reactants are lower energetically than the products. As seen in Fig. 8 the reactants have a lower potential energy than the products.

=== Locating the TS ===

[[File:mc_TS_exer2.PNG|thumb|center|800px|'''Fig. 10''' Internuclear Distance against time and forces in TS for the reaction F + H2]]
According to Hammond’s Postulate the TS in a reaction resembles either the reactants or products depending on where it energetically is: an early transition state is energetically closer to the reactants, whereas a late TS is energetically closer products. Early transition states are characteristic of exothermic reactions such as the H2 + F -> HF + H reaction. Therefore, the TS must resemble the reactants, specifically H2. The H2 bond length was 0.74 Å and then decimal places were added to both the HF and H2 until the the position of the TS was found to be HF=0.744878 Å and H2=1.810789 Å. The KE and the forces along the AB and BC being 0 confirmed the TS position.

=== Activation Energy ===
The activation energy was found by calculating the energetic difference between the TS and the reactants when the reacting species are no longer in the TS. This was achieved by reducing/increasing the value of rAB and rBC. The activation energy was then calculated to be:
[[File:mc_AE_Reaction1.PNG|thumb|center|500px|'''Fig. 11''' Energy against Time plot for the F + H2 to find the Ea]]
Ea of H2 + F reaction=0.275 kcal/mol
[[File:mc_AE_Reaction2.PNG |thumb|center|500px|'''Fig. 12''' Energy against Time plot for the H + HF to find the Ea]]
Ea of HF + H reaction=28.25 kcal/mol

===Reaction dynamics: F + H2 → HF + H===

To achieve a reaction trajectory the conditions were set to:
<li>rHF = 1.79 Å
<li>rHH = 0.74 Å
<li>pHH = pHF = 0

[[File: mc_momentum_energyplot.PNG|thumb|center|800px|Left:'''Fig. 13''' Momentum against time plot for the reaction F + H2 Right:'''Fig. 14''' Energy against time plot for the reaction F + H2]]
Initially the H2 molecule travels straight with low amplitude oscillations as seen in Figure 12. Following the collision of H2 with the F atom the produced HF molecule has excess KE due to the exothermic reaction (formation of HF bond) and is vibrationally excited, oscillating at high amplitudes. In light of the fact that energy is conserved the total energy in this reaction system remains constant as seen in Figure 12. Instead, PE is converted to KE as the atoms collude manifested in the HF oscillations.
 This can be experimentally observed form IR-emission spectroscopy. Following the formation of the products, the HF bond is vibrationally excited therefore, the IR emitted of the HF can be detected. (ref 10 ming)

===Polanyi's Empirical Rules===
In chemical reactions, in order for the reactants to become products they have to overcome the activation energy barrier. The Polyani Rules explain that vibrational energy promotes a late-barrier reaction (endothermic–product resembling TS) more efficiently, in contrast to translational energy that promotes early-barrier reaction (exothermic–reactant resembling TS).<ref name="ref4" />
 The forward reaction H2 + F is exothermic, as stated above. Therefore, it is an early-barrier reaction and the TS is energetically closer to the reactants. As per Polyani’s Rules, translational energy is more efficient in this reaction.
 The reverse reaction HF + H is endothermic, as stated above. Therefore, it is a late-barrier reaction and the TS is energetically closer to the products. As per Polyani’s Rules, vibrational energy is more efficient in this reaction.

===Forward Reaction: H2 + F → HF + H===
====Test 1====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= -3.0 Å
<li>pHF= -0.5 Å

[[File: mc_experiment1_exo.PNG|thumb|center|800px|'''Fig. 15''' Contour plot for the reaction F + H2 in Test 1]]
As seen in Figure 15 the reaction does not go to completion in these experimental conditions.
====Test 2====
System Conditions:
<li> rHH= 0.74 Å
<li>rHF= 1.8 Å
<li>pHH= 0.1 Å
<li>pHF= -0.8 Å
[[File: mc_experiment2_exo.PNG|thumb|center|800px|'''Fig. 16''' Contour plot for the reaction F + H2 in Test 2]]
Reaction goes to completion

===Reverse Reaction: HF + H → H2 + F===

====Test 1====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= 0.05 Å
<li>pHF= -3.0 Å

[[File: mc_experiment1_endo.PNG|thumb|center|800px|'''Fig. 17''' Contour plot for the reaction HF + H in Test 1]]
Reaction does not go to completion
====Test 2====
System Conditions:
<li> rHH= 2.0 Å
<li>rHF= 0.92 Å
<li>pHH= -0.3 Å
<li>pHF= 7.8 Å

[[File: mc_experiment2_endo.PNG|thumb|center|800px|'''Fig. 18''' Contour plot for the reaction HF + H in Test 2]]

MRD:mc4717

2019-05-16T15:24:18Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */

MRD:mc4717

2019-05-16T15:23:58Z

Mc4717: /* Activation Energy */

MRD:mc4717

2019-05-16T15:19:31Z

Mc4717: /* Reverse Reaction: HF + H → H2 + F */

MRD:mc4717

2019-05-16T15:18:04Z

Mc4717: /* Test 2 */

MRD:mc4717

2019-05-16T15:16:02Z

Mc4717: /* Forward Reaction: H2 + F → HF + H */

File:Mc experiment2 exo.PNG

2019-05-16T15:01:57Z

Mc4717:

File:Mc experiment2 endo.PNG

2019-05-16T15:01:46Z

Mc4717:

File:Mc experiment1 exo.PNG

2019-05-16T15:01:34Z

Mc4717:

File:Mc experiment1 endo.PNG

2019-05-16T15:01:22Z

Mc4717:

MRD:mc4717

2019-05-16T14:14:47Z

Mc4717: /* HF + H → H2 + F */

MRD:mc4717

2019-05-16T14:14:28Z

Mc4717: /* Polanyi's Empirical Rules */

MRD:mc4717

2019-05-16T14:12:00Z

Mc4717: /* Polanyi's Empirical Rules */

MRD:mc4717

2019-05-16T14:09:38Z

Mc4717: /* Polanyi's Empirical Rules */

MRD:mc4717

2019-05-16T13:51:58Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */

MRD:mc4717

2019-05-16T13:48:29Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */

MRD:mc4717

2019-05-16T13:48:16Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */

MRD:mc4717

2019-05-16T13:42:21Z

Mc4717: /* Reaction dynamics: F + H2 → HF + H */