File:Yg8818 h2 ae.png

2020-05-08T15:54:47Z

Yg8818:

2020-05-08T14:51:29Z

Yg8818:

File:Yg8818 early ts HF.png

2020-05-08T14:51:11Z

Yg8818:

MRD:01533336yg8818

2020-05-08T14:10:42Z

Yg8818:

== Molecular Reaction Dynamics Lab Report ==

== H and H2 System ==

The transition state of a reaction is defined as a point on the minimum energy path between reactants and products where the derivatives in all orthogonal directions are zero. Mathematically, the transition state, or saddle point, is recognised by computing the function's Hessian matrix at that point. If the Hessian matrix is indefinite, the point must be a saddle point. The saddle point can be distinguished from an energy minimum point, as ∂2V/ ∂r1 > 0 and ∂2V/ ∂r2 > 0 in both orthogonal directions of a minimum point, indicating a stable region of reactants or products.

The reaction studied here is one in which a hydrogen atom collides with a hydrogen molecule to form a new hydrogen molecule plus a hydrogen atom. This is one of the simplest systems to study, as the bimolecular collision contains only three protons and three electrons. r1 denotes the distance between atom 1 and atom 2 and r2 is the distance between atom 2 and atom 3.

The transition state of the reaction between a single hydrogen atom, H, with diatomic hydrogen, H2, must occur when the interatomic distances are equal. This is because the reaction is neither exothermic or endothermic in nature. The transition state must not be late or early, but directly equidistant between reactants and products. As a result, r1 must be equal to r2 and their momenta zero as well. With no oscillation, the gradient, force and acceleration of the particles must also be zero at the transition state.

[[File:Yg8818 int dist hhh.png|left|250 px|thumb|Figure 1ː Interatomic Distances vs Time plot for the H and H2 model]]

When r1 = r2 = 90.775 pm, the above conditions are met and the transition state is observed. This is visible as a single point on the surface plot and as straight lines with no oscillations on the Interatomic Distances vs Time plot, as shown in Figure 1 (left).

[[File:Yg8818 MEP hhh.png|250 px|thumb|Figure 2ː MEP for the H and H2 model]]
[[File:Yg8818 dynamics hhh.png|250 px|thumb|Figure 3ː Dynamics trajectory for the H and H2 model]]

Now that the transition state has been located, the reaction path, which is the Minimum Energy Path (MEP), can be determined. This is a trajectory that corresponds to infinitely slow motion of the particles and can be found by slightly altering the value of one of the interatomic distances. Here, r1 is set equal to 91.775 pm, just 1 pm higher than used for finding the transition state. This trajectory is modelled in Figure 2 (right).

The MEP doesn't account for mass and inertial motion of the atoms. While this provides a useful simplification, the model does not accurately reflect how the particles move during the reaction. This can be resolved by maintaining the current parameters, but using a dynamic calculation type for the interatomic distances vs. time plot. This is displayed in Figure 3 (right). Altering the distance for r2 instead of r1 would yield the same results.

Trajectories with initial conditions in the range r1 = 74 pm, r2 = 200 pm, with -3.1 < p1/ g.mol-1.pm.fs-1 < -1.6 and p2 = -5.1 g.mol-1.pm.fs-1 have shown to be reactive in the above calculations. Here, higher values of momenta have been trialled, while keeping the position values the same, in determining whether the species will still be reactive and the appearance of the trajectory. The results are displayed in Table 1 below.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.280 || style="text-align: center;" |Yes || The atoms here approach the transition state with no oscillation at all. Having crossed the saddle point, the resultant products begin to oscillate noticeably. || [[File:Yg8818_tp1.png| 250px]]
|-
| -3.1 || -4.1 || -420.077 || style="text-align: center;" |No || No reaction occurs at these momenta values. It is evident from the contour plot that the molecules approach the transition state while oscillating, but don't have enough kinetic energy to pass over the saddle point. The reactants then return to their original state. || [[File:Yg8818_tp2.png| 250px]]
|-
| -3.1 || -5.1 || -413.977 || style="text-align: center;" |Yes || With an increased amount of momentum for p2, the reactants reach the saddle point without oscillation. The route from transition state to products is full of high energy oscillation (greater than for the first reaction outlined in the table) which can be directly related to the molecules having more kinetic energy. || [[File:Yg8818_tp3.png| 250px]]
|-
| -5.1 || -10.1 || -357.277 || style="text-align: center;" |Yes || After the transition state has been surmounted and products are forming, lots of oscillations can be observed on this contour plot. This is as a result of yet another increase in kinetic energy, although the total energy (which accounts for potential energy as well) is lower than for some of the other examples. || [[File:Yg8818_tp4.png| 250px]]
|-
| -5.1 || -10.6 || -349.477 || style="text-align: center;" |Yes || These conditions display an example of barrier recrossing. Here, transition state is reach by the reactants, but their extremely high kinetic energy results in strong oscillations. These prevent the molecules forming products and instead they revert back to their original state. This occurrence cannot be explained by transition state theory, as ordinarily, once a reaction has passed the saddle point it should progress to form products. || [[File:Yg8818_tp5.png| 250px]]
|}

It can be inferred from Table 1 that an increase in potential (and hence kinetic energy), for either distance, results in molecules with greater oscillation. If the potential energy is too low, transition state won't be reached. If the kinetic energy component is too large, oscillations can interfere with the reaction, resulting in a return to reactants even after the saddle point has been crested. A balance found between the two is essential for reaction to occur.

These calculations are all theoretical and based entirely on transition state theory. Transition state theory complies with several key assumptionsː
# The energy of the particles in the reaction follows the Boltzmann distribution
# The reactants are in constant equilibrium with the transition state structure
# Effects of quantum tunnelling are negligible
# Motion along the reaction coordinate can be separated from other motions and treated classically at the transition state
# Once the transition state has been achieved, the reactants will not reform

This final assumption has shown to be untrue in some circumstances, as in the last example studied (Table 1). Transition state theory does not account for barrier recrossing which is observed frequently in experiment. This could result in transition state theory predicting reactions to have a faster rate than experiment, as a slower rate would be observed with barrier recrossing.

Ignoring effects of quantum tunnelling (as mention above) assumes that the reaction will not take place unless sufficient activation energy is reached to overcome the transition state. However, according to quantum mechanics, particles can still tunnel across the barrier, if the barrier is of finite energy. In experiment, molecules will react sometimes, even if they do not collide with enough energy to scale the transition state barrier. This is an important feature for reactions with small activation energies (tunnelling probability decreases with increasing barrier height) and should not be ignored. Transition state theory will therefore predict that some reactions won't occur, while they may occur successfully in experiment.

In its calculation of reaction rate, transition state theory is limited for reactions occurring at high temperatures. The theory assumes that the system will traverse the lowest energy saddle point on the potential energy surface, which is not always true for systems at high temperatures. At high temperatures, molecules will populate higher energy modes and transitions states will be crosses that are far from the lowest energy saddle point. This can even be observed in this very collision of H with H2 when the experiment is conducted at increased temperatures.

A combination of all of these factors results in transition state theory falling short, in some cases, in its prediction of reaction rate.

== F - H - H System ==

Inspection of the potential energy surfaces for the reactions of F with H - H and H with F - H can be used to classify their energetics; determining if they are exothermic or endothermic reactions. Exothermic reactions have 'early' transition states, as the structure of the transition state closely resembles that of the reactants. Endothermic reactions have 'late' transition states, as the structure of the transition state closely resembles that of the products. This is based on Hammond's Postulate which asserts that if two states (eg. a transition state and an intermediate) occur consecutively in a reaction and have nearly the same energy, their interconversion will only involve a small reorganisation of molecular structure.

The reaction of F with H - H has an early transition state (Figure 4) rendering it an exothermic reaction; the transition state close in position to the reactant bond lengths. Meanwhile, when H collides with F - H, a late transition state is observed (Figure 5) indicating an endothermic reaction where the saddle point lies close to the product bond lengths. This can be explained by the bond strength of H - F, which takes a value of 565 KJ mol-1. This is much greater than the bond energy of H - H which is 432 KJ mol-1. More energy is required to break the H - F bond than is released in forming the H - H bond.

[[File:Yg8818_exo_ts_fhh.png|300 px|thumb|Figure 4ː Reaction of F with H2. The early transition state is visible, closer to the initial reactant bond lengths.]]

The transition state is found when H - F = 181.000 pm and H - H =74.487 pm, with momenta set to zero for both. The Internuclear Distance vs Time graph (Figure 6) illustrates this well, as a straight line on the plot with no oscillations, characteristic of a transition state.

[[File:Yg8818_ts_fhh.png|400 px|thumb|Figure 6ː Transition state for the F - H - H model]]

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File:Yg8818 exo ts fhh.png

2020-05-08T14:02:50Z

Yg8818:

MRD:01533336yg8818

2020-05-08T13:59:31Z

Yg8818: /* F - H - H System */

== Molecular Reaction Dynamics Lab Report ==

== H and H2 System ==

The transition state of a reaction is defined as a point on the minimum energy path between reactants and products where the derivatives in all orthogonal directions are zero. Mathematically, the transition state, or saddle point, is recognised by computing the function's Hessian matrix at that point. If the Hessian matrix is indefinite, the point must be a saddle point. The saddle point can be distinguished from an energy minimum point, as ∂2V/ ∂r1 > 0 and ∂2V/ ∂r2 > 0 in both orthogonal directions of a minimum point, indicating a stable region of reactants or products.

The reaction studied here is one in which a hydrogen atom collides with a hydrogen molecule to form a new hydrogen molecule plus a hydrogen atom. This is one of the simplest systems to study, as the bimolecular collision contains only three protons and three electrons. r1 denotes the distance between atom 1 and atom 2 and r2 is the distance between atom 2 and atom 3.

The transition state of the reaction between a single hydrogen atom, H, with diatomic hydrogen, H2, must occur when the interatomic distances are equal. This is because the reaction is neither exothermic or endothermic in nature. The transition state must not be late or early, but directly equidistant between reactants and products. As a result, r1 must be equal to r2 and their momenta zero as well. With no oscillation, the gradient, force and acceleration of the particles must also be zero at the transition state.

[[File:Yg8818 int dist hhh.png|left|250 px|thumb|Figure 1ː Interatomic Distances vs Time plot for the H and H2 model]]

When r1 = r2 = 90.775 pm, the above conditions are met and the transition state is observed. This is visible as a single point on the surface plot and as straight lines with no oscillations on the Interatomic Distances vs Time plot, as shown in Figure 1 (left).

[[File:Yg8818 MEP hhh.png|250 px|thumb|Figure 2ː MEP for the H and H2 model]]
[[File:Yg8818 dynamics hhh.png|250 px|thumb|Figure 3ː Dynamics trajectory for the H and H2 model]]

Now that the transition state has been located, the reaction path, which is the Minimum Energy Path (MEP), can be determined. This is a trajectory that corresponds to infinitely slow motion of the particles and can be found by slightly altering the value of one of the interatomic distances. Here, r1 is set equal to 91.775 pm, just 1 pm higher than used for finding the transition state. This trajectory is modelled in Figure 2 (right).

The MEP doesn't account for mass and inertial motion of the atoms. While this provides a useful simplification, the model does not accurately reflect how the particles move during the reaction. This can be resolved by maintaining the current parameters, but using a dynamic calculation type for the interatomic distances vs. time plot. This is displayed in Figure 3 (right). Altering the distance for r2 instead of r1 would yield the same results.

Trajectories with initial conditions in the range r1 = 74 pm, r2 = 200 pm, with -3.1 < p1/ g.mol-1.pm.fs-1 < -1.6 and p2 = -5.1 g.mol-1.pm.fs-1 have shown to be reactive in the above calculations. Here, higher values of momenta have been trialled, while keeping the position values the same, in determining whether the species will still be reactive and the appearance of the trajectory. The results are displayed in Table 1 below.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.280 || style="text-align: center;" |Yes || The atoms here approach the transition state with no oscillation at all. Having crossed the saddle point, the resultant products begin to oscillate noticeably. || [[File:Yg8818_tp1.png| 250px]]
|-
| -3.1 || -4.1 || -420.077 || style="text-align: center;" |No || No reaction occurs at these momenta values. It is evident from the contour plot that the molecules approach the transition state while oscillating, but don't have enough kinetic energy to pass over the saddle point. The reactants then return to their original state. || [[File:Yg8818_tp2.png| 250px]]
|-
| -3.1 || -5.1 || -413.977 || style="text-align: center;" |Yes || With an increased amount of momentum for p2, the reactants reach the saddle point without oscillation. The route from transition state to products is full of high energy oscillation (greater than for the first reaction outlined in the table) which can be directly related to the molecules having more kinetic energy. || [[File:Yg8818_tp3.png| 250px]]
|-
| -5.1 || -10.1 || -357.277 || style="text-align: center;" |Yes || After the transition state has been surmounted and products are forming, lots of oscillations can be observed on this contour plot. This is as a result of yet another increase in kinetic energy, although the total energy (which accounts for potential energy as well) is lower than for some of the other examples. || [[File:Yg8818_tp4.png| 250px]]
|-
| -5.1 || -10.6 || -349.477 || style="text-align: center;" |Yes || These conditions display an example of barrier recrossing. Here, transition state is reach by the reactants, but their extremely high kinetic energy results in strong oscillations. These prevent the molecules forming products and instead they revert back to their original state. This occurrence cannot be explained by transition state theory, as ordinarily, once a reaction has passed the saddle point it should progress to form products. || [[File:Yg8818_tp5.png| 250px]]
|}

It can be inferred from Table 1 that an increase in potential (and hence kinetic energy), for either distance, results in molecules with greater oscillation. If the potential energy is too low, transition state won't be reached. If the kinetic energy component is too large, oscillations can interfere with the reaction, resulting in a return to reactants even after the saddle point has been crested. A balance found between the two is essential for reaction to occur.

These calculations are all theoretical and based entirely on transition state theory. Transition state theory complies with several key assumptionsː
# The energy of the particles in the reaction follows the Boltzmann distribution
# The reactants are in constant equilibrium with the transition state structure
# Effects of quantum tunnelling are negligible
# Motion along the reaction coordinate can be separated from other motions and treated classically at the transition state
# Once the transition state has been achieved, the reactants will not reform

This final assumption has shown to be untrue in some circumstances, as in the last example studied (Table 1). Transition state theory does not account for barrier recrossing which is observed frequently in experiment. This could result in transition state theory predicting reactions to have a faster rate than experiment, as a slower rate would be observed with barrier recrossing.

Ignoring effects of quantum tunnelling (as mention above) assumes that the reaction will not take place unless sufficient activation energy is reached to overcome the transition state. However, according to quantum mechanics, particles can still tunnel across the barrier, if the barrier is of finite energy. In experiment, molecules will react sometimes, even if they do not collide with enough energy to scale the transition state barrier. This is an important feature for reactions with small activation energies (tunnelling probability decreases with increasing barrier height) and should not be ignored. Transition state theory will therefore predict that some reactions won't occur, while they may occur successfully in experiment.

In its calculation of reaction rate, transition state theory is limited for reactions occurring at high temperatures. The theory assumes that the system will traverse the lowest energy saddle point on the potential energy surface, which is not always true for systems at high temperatures. At high temperatures, molecules will populate higher energy modes and transitions states will be crosses that are far from the lowest energy saddle point. This can even be observed in this very collision of H with H2 when the experiment is conducted at increased temperatures.

A combination of all of these factors results in transition state theory falling short, in some cases, in its prediction of reaction rate.

== F - H - H System ==

Inspection of the potential energy surfaces for the reactions of F with H - H and H with F - H can be used to classify their energetics; determining if they are exothermic or endothermic reactions. Exothermic reactions have 'early' transition states, as the structure of the transition state closely resembles that of the reactants. Endothermic reactions have 'late' transition states, as the structure of the transition state closely resembles that of the products. This is based on Hammond's Postulate which asserts that if two states (eg. a transition state and an intermediate) occur consecutively in a reaction and have nearly the same energy, their interconversion will only involve a small reorganisation of molecular structure.

The reaction of F with H - H has an early transition state (Figure 4) rendering it an exothermic reaction; the transition state close in position to the reactant bond lengths. Meanwhile, when H collides with F - H, a late transition state is observed (Figure 5) indicating an endothermic reaction where the saddle point lies close to the product bond lengths. This can be explained by the bond strength of H - F, which takes a value of 565 KJ mol-1. This is much greater than the bond energy of H - H which is 432 KJ mol-1. More energy is required to break the H - F bond than is released in forming the H - H bond.

The transition state is found when H - F = 181.000 pm and H - H =74.487 pm, with momenta set to zero for both. The Internuclear Distance vs Time graph (Figure 6) illustrates this well, as a straight line on the plot with no oscillations, characteristic of a transition state.

[[File:Yg8818_ts_fhh.png|400 px|thumb|Figure 6ː Transition state for the F - H - H model]]

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2020-05-08T13:43:57Z

Yg8818: /* F - H - H System */

== Molecular Reaction Dynamics Lab Report ==

== H and H2 System ==

The transition state of a reaction is defined as a point on the minimum energy path between reactants and products where the derivatives in all orthogonal directions are zero. Mathematically, the transition state, or saddle point, is recognised by computing the function's Hessian matrix at that point. If the Hessian matrix is indefinite, the point must be a saddle point. The saddle point can be distinguished from an energy minimum point, as ∂2V/ ∂r1 > 0 and ∂2V/ ∂r2 > 0 in both orthogonal directions of a minimum point, indicating a stable region of reactants or products.

The reaction studied here is one in which a hydrogen atom collides with a hydrogen molecule to form a new hydrogen molecule plus a hydrogen atom. This is one of the simplest systems to study, as the bimolecular collision contains only three protons and three electrons. r1 denotes the distance between atom 1 and atom 2 and r2 is the distance between atom 2 and atom 3.

The transition state of the reaction between a single hydrogen atom, H, with diatomic hydrogen, H2, must occur when the interatomic distances are equal. This is because the reaction is neither exothermic or endothermic in nature. The transition state must not be late or early, but directly equidistant between reactants and products. As a result, r1 must be equal to r2 and their momenta zero as well. With no oscillation, the gradient, force and acceleration of the particles must also be zero at the transition state.

[[File:Yg8818 int dist hhh.png|left|250 px|thumb|Figure 1ː Interatomic Distances vs Time plot for the H and H2 model]]

When r1 = r2 = 90.775 pm, the above conditions are met and the transition state is observed. This is visible as a single point on the surface plot and as straight lines with no oscillations on the Interatomic Distances vs Time plot, as shown in Figure 1 (left).

[[File:Yg8818 MEP hhh.png|250 px|thumb|Figure 2ː MEP for the H and H2 model]]
[[File:Yg8818 dynamics hhh.png|250 px|thumb|Figure 3ː Dynamics trajectory for the H and H2 model]]

Now that the transition state has been located, the reaction path, which is the Minimum Energy Path (MEP), can be determined. This is a trajectory that corresponds to infinitely slow motion of the particles and can be found by slightly altering the value of one of the interatomic distances. Here, r1 is set equal to 91.775 pm, just 1 pm higher than used for finding the transition state. This trajectory is modelled in Figure 2 (right).

The MEP doesn't account for mass and inertial motion of the atoms. While this provides a useful simplification, the model does not accurately reflect how the particles move during the reaction. This can be resolved by maintaining the current parameters, but using a dynamic calculation type for the interatomic distances vs. time plot. This is displayed in Figure 3 (right). Altering the distance for r2 instead of r1 would yield the same results.

Trajectories with initial conditions in the range r1 = 74 pm, r2 = 200 pm, with -3.1 < p1/ g.mol-1.pm.fs-1 < -1.6 and p2 = -5.1 g.mol-1.pm.fs-1 have shown to be reactive in the above calculations. Here, higher values of momenta have been trialled, while keeping the position values the same, in determining whether the species will still be reactive and the appearance of the trajectory. The results are displayed in Table 1 below.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.280 || style="text-align: center;" |Yes || The atoms here approach the transition state with no oscillation at all. Having crossed the saddle point, the resultant products begin to oscillate noticeably. || [[File:Yg8818_tp1.png| 250px]]
|-
| -3.1 || -4.1 || -420.077 || style="text-align: center;" |No || No reaction occurs at these momenta values. It is evident from the contour plot that the molecules approach the transition state while oscillating, but don't have enough kinetic energy to pass over the saddle point. The reactants then return to their original state. || [[File:Yg8818_tp2.png| 250px]]
|-
| -3.1 || -5.1 || -413.977 || style="text-align: center;" |Yes || With an increased amount of momentum for p2, the reactants reach the saddle point without oscillation. The route from transition state to products is full of high energy oscillation (greater than for the first reaction outlined in the table) which can be directly related to the molecules having more kinetic energy. || [[File:Yg8818_tp3.png| 250px]]
|-
| -5.1 || -10.1 || -357.277 || style="text-align: center;" |Yes || After the transition state has been surmounted and products are forming, lots of oscillations can be observed on this contour plot. This is as a result of yet another increase in kinetic energy, although the total energy (which accounts for potential energy as well) is lower than for some of the other examples. || [[File:Yg8818_tp4.png| 250px]]
|-
| -5.1 || -10.6 || -349.477 || style="text-align: center;" |Yes || These conditions display an example of barrier recrossing. Here, transition state is reach by the reactants, but their extremely high kinetic energy results in strong oscillations. These prevent the molecules forming products and instead they revert back to their original state. This occurrence cannot be explained by transition state theory, as ordinarily, once a reaction has passed the saddle point it should progress to form products. || [[File:Yg8818_tp5.png| 250px]]
|}

It can be inferred from Table 1 that an increase in potential (and hence kinetic energy), for either distance, results in molecules with greater oscillation. If the potential energy is too low, transition state won't be reached. If the kinetic energy component is too large, oscillations can interfere with the reaction, resulting in a return to reactants even after the saddle point has been crested. A balance found between the two is essential for reaction to occur.

These calculations are all theoretical and based entirely on transition state theory. Transition state theory complies with several key assumptionsː
# The energy of the particles in the reaction follows the Boltzmann distribution
# The reactants are in constant equilibrium with the transition state structure
# Effects of quantum tunnelling are negligible
# Motion along the reaction coordinate can be separated from other motions and treated classically at the transition state
# Once the transition state has been achieved, the reactants will not reform

This final assumption has shown to be untrue in some circumstances, as in the last example studied (Table 1). Transition state theory does not account for barrier recrossing which is observed frequently in experiment. This could result in transition state theory predicting reactions to have a faster rate than experiment, as a slower rate would be observed with barrier recrossing.

Ignoring effects of quantum tunnelling (as mention above) assumes that the reaction will not take place unless sufficient activation energy is reached to overcome the transition state. However, according to quantum mechanics, particles can still tunnel across the barrier, if the barrier is of finite energy. In experiment, molecules will react sometimes, even if they do not collide with enough energy to scale the transition state barrier. This is an important feature for reactions with small activation energies (tunnelling probability decreases with increasing barrier height) and should not be ignored. Transition state theory will therefore predict that some reactions won't occur, while they may occur successfully in experiment.

In its calculation of reaction rate, transition state theory is limited for reactions occurring at high temperatures. The theory assumes that the system will traverse the lowest energy saddle point on the potential energy surface, which is not always true for systems at high temperatures. At high temperatures, molecules will populate higher energy modes and transitions states will be crosses that are far from the lowest energy saddle point. This can even be observed in this very collision of H with H2 when the experiment is conducted at increased temperatures.

A combination of all of these factors results in transition state theory falling short, in some cases, in its prediction of reaction rate.

== F - H - H System ==

Inspection of the potential energy surfaces for the reactions of F with H - H and H with F - H (Figures 4 and 5) can be used to classify their energetics; determining if they are exothermic or endothermic reactions. Exothermic reactions have 'early' transition states, as the structure of the transition state closely resembles that of the reactants. Endothermic reactions have 'late' transition states, as the structure of the transition state closely resembles that of the products. This is based on Hammond's Postulate which asserts that if two states (eg. a transition state and an intermediate) occur consecutively in a reaction and have nearly the same energy, their interconversion will only involve a small reorganisation of molecular structure.

The reaction of F with H - H has an early transition state, rendering it an exothermic reaction; the transition state close in position to the reactant bond lengths. Meanwhile, when H collides with F - H, a late transition state is observed indicating an endothermic reaction where the saddle point lies close to the product bond lengths. This can be explained by the bond strength of H - F, which takes a value of 565 KJ mol-1. This is much greater than the bond energy of H - H which is 432 KJ mol-1. More energy is required to break the H - F bond than is released in forming the H - H bond.

The transition state is found when H - F = 181.000 pm and H - H =74.4870 pm, with momenta set to zero for both. The Internuclear Distance vs Time graph (Figure 6) illustrates this well, as a straight line on the plot with no oscillations, characteristic of a transition state.

[[File:Yg8818_ts_fhh.png|400 px|thumb|Figure 6ː Transition state for the F - H - H model]]

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2020-05-08T13:42:42Z

Yg8818:

== Molecular Reaction Dynamics Lab Report ==

== H and H2 System ==

The transition state of a reaction is defined as a point on the minimum energy path between reactants and products where the derivatives in all orthogonal directions are zero. Mathematically, the transition state, or saddle point, is recognised by computing the function's Hessian matrix at that point. If the Hessian matrix is indefinite, the point must be a saddle point. The saddle point can be distinguished from an energy minimum point, as ∂2V/ ∂r1 > 0 and ∂2V/ ∂r2 > 0 in both orthogonal directions of a minimum point, indicating a stable region of reactants or products.

The reaction studied here is one in which a hydrogen atom collides with a hydrogen molecule to form a new hydrogen molecule plus a hydrogen atom. This is one of the simplest systems to study, as the bimolecular collision contains only three protons and three electrons. r1 denotes the distance between atom 1 and atom 2 and r2 is the distance between atom 2 and atom 3.

The transition state of the reaction between a single hydrogen atom, H, with diatomic hydrogen, H2, must occur when the interatomic distances are equal. This is because the reaction is neither exothermic or endothermic in nature. The transition state must not be late or early, but directly equidistant between reactants and products. As a result, r1 must be equal to r2 and their momenta zero as well. With no oscillation, the gradient, force and acceleration of the particles must also be zero at the transition state.

[[File:Yg8818 int dist hhh.png|left|250 px|thumb|Figure 1ː Interatomic Distances vs Time plot for the H and H2 model]]

When r1 = r2 = 90.775 pm, the above conditions are met and the transition state is observed. This is visible as a single point on the surface plot and as straight lines with no oscillations on the Interatomic Distances vs Time plot, as shown in Figure 1 (left).

[[File:Yg8818 MEP hhh.png|250 px|thumb|Figure 2ː MEP for the H and H2 model]]
[[File:Yg8818 dynamics hhh.png|250 px|thumb|Figure 3ː Dynamics trajectory for the H and H2 model]]

Now that the transition state has been located, the reaction path, which is the Minimum Energy Path (MEP), can be determined. This is a trajectory that corresponds to infinitely slow motion of the particles and can be found by slightly altering the value of one of the interatomic distances. Here, r1 is set equal to 91.775 pm, just 1 pm higher than used for finding the transition state. This trajectory is modelled in Figure 2 (right).

The MEP doesn't account for mass and inertial motion of the atoms. While this provides a useful simplification, the model does not accurately reflect how the particles move during the reaction. This can be resolved by maintaining the current parameters, but using a dynamic calculation type for the interatomic distances vs. time plot. This is displayed in Figure 3 (right). Altering the distance for r2 instead of r1 would yield the same results.

Trajectories with initial conditions in the range r1 = 74 pm, r2 = 200 pm, with -3.1 < p1/ g.mol-1.pm.fs-1 < -1.6 and p2 = -5.1 g.mol-1.pm.fs-1 have shown to be reactive in the above calculations. Here, higher values of momenta have been trialled, while keeping the position values the same, in determining whether the species will still be reactive and the appearance of the trajectory. The results are displayed in Table 1 below.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.280 || style="text-align: center;" |Yes || The atoms here approach the transition state with no oscillation at all. Having crossed the saddle point, the resultant products begin to oscillate noticeably. || [[File:Yg8818_tp1.png| 250px]]
|-
| -3.1 || -4.1 || -420.077 || style="text-align: center;" |No || No reaction occurs at these momenta values. It is evident from the contour plot that the molecules approach the transition state while oscillating, but don't have enough kinetic energy to pass over the saddle point. The reactants then return to their original state. || [[File:Yg8818_tp2.png| 250px]]
|-
| -3.1 || -5.1 || -413.977 || style="text-align: center;" |Yes || With an increased amount of momentum for p2, the reactants reach the saddle point without oscillation. The route from transition state to products is full of high energy oscillation (greater than for the first reaction outlined in the table) which can be directly related to the molecules having more kinetic energy. || [[File:Yg8818_tp3.png| 250px]]
|-
| -5.1 || -10.1 || -357.277 || style="text-align: center;" |Yes || After the transition state has been surmounted and products are forming, lots of oscillations can be observed on this contour plot. This is as a result of yet another increase in kinetic energy, although the total energy (which accounts for potential energy as well) is lower than for some of the other examples. || [[File:Yg8818_tp4.png| 250px]]
|-
| -5.1 || -10.6 || -349.477 || style="text-align: center;" |Yes || These conditions display an example of barrier recrossing. Here, transition state is reach by the reactants, but their extremely high kinetic energy results in strong oscillations. These prevent the molecules forming products and instead they revert back to their original state. This occurrence cannot be explained by transition state theory, as ordinarily, once a reaction has passed the saddle point it should progress to form products. || [[File:Yg8818_tp5.png| 250px]]
|}

It can be inferred from Table 1 that an increase in potential (and hence kinetic energy), for either distance, results in molecules with greater oscillation. If the potential energy is too low, transition state won't be reached. If the kinetic energy component is too large, oscillations can interfere with the reaction, resulting in a return to reactants even after the saddle point has been crested. A balance found between the two is essential for reaction to occur.

These calculations are all theoretical and based entirely on transition state theory. Transition state theory complies with several key assumptionsː
# The energy of the particles in the reaction follows the Boltzmann distribution
# The reactants are in constant equilibrium with the transition state structure
# Effects of quantum tunnelling are negligible
# Motion along the reaction coordinate can be separated from other motions and treated classically at the transition state
# Once the transition state has been achieved, the reactants will not reform

This final assumption has shown to be untrue in some circumstances, as in the last example studied (Table 1). Transition state theory does not account for barrier recrossing which is observed frequently in experiment. This could result in transition state theory predicting reactions to have a faster rate than experiment, as a slower rate would be observed with barrier recrossing.

Ignoring effects of quantum tunnelling (as mention above) assumes that the reaction will not take place unless sufficient activation energy is reached to overcome the transition state. However, according to quantum mechanics, particles can still tunnel across the barrier, if the barrier is of finite energy. In experiment, molecules will react sometimes, even if they do not collide with enough energy to scale the transition state barrier. This is an important feature for reactions with small activation energies (tunnelling probability decreases with increasing barrier height) and should not be ignored. Transition state theory will therefore predict that some reactions won't occur, while they may occur successfully in experiment.

In its calculation of reaction rate, transition state theory is limited for reactions occurring at high temperatures. The theory assumes that the system will traverse the lowest energy saddle point on the potential energy surface, which is not always true for systems at high temperatures. At high temperatures, molecules will populate higher energy modes and transitions states will be crosses that are far from the lowest energy saddle point. This can even be observed in this very collision of H with H2 when the experiment is conducted at increased temperatures.

A combination of all of these factors results in transition state theory falling short, in some cases, in its prediction of reaction rate.

== F - H - H System ==

Inspection of the potential energy surfaces for the reactions of F with H - H and H with F - H can be used to classify their energetics; determining if they are exothermic or endothermic reactions. Exothermic reactions have 'early' transition states, as the structure of the transition state closely resembles that of the reactants. Endothermic reactions have 'late' transition states, as the structure of the transition state closely resembles that of the products. This is based on Hammond's Postulate which asserts that if two states (eg. a transition state and an intermediate) occur consecutively in a reaction and have nearly the same energy, their interconversion will only involve a small reorganisation of molecular structure.

The reaction of F with H - H has an early transition state, rendering it an exothermic reaction; the transition state close in position to the reactant bond lengths. Meanwhile, when H collides with F - H, a late transition state is observed indicating an endothermic reaction where the saddle point lies close to the product bond lengths. This can be explained by the bond strength of H - F, which takes a value of 565 KJ mol-1. This is much greater than the bond energy of H - H which is 432 KJ mol-1. More energy is required to break the H - F bond than is released in forming the H - H bond.

The transition state is found when H - F = 181.000 pm and H - H =74.4870 pm, with momenta set to zero for both. The Internuclear Distance vs Time graph (Figure 4) illustrates this well, as a straight line on the plot with no oscillations, characteristic of a transition state.

[[File:Yg8818_ts_fhh.png|400 px|thumb|Figure 4ː Transition state for the F - H - H model]]

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Yg8818: /* F - H - H System */

== Molecular Reaction Dynamics Lab Report ==

== H and H2 System ==

The transition state of a reaction is defined as a point on the minimum energy path between reactants and products where the derivatives in all orthogonal directions are zero. Mathematically, the transition state, or saddle point, is recognised by computing the function's Hessian matrix at that point. If the Hessian matrix is indefinite, the point must be a saddle point. The saddle point can be distinguished from an energy minimum point, as ∂2V/ ∂r1 > 0 and ∂2V/ ∂r2 > 0 in both orthogonal directions of a minimum point, indicating a stable region of reactants or products.

The reaction studied here is one in which a hydrogen atom collides with a hydrogen molecule to form a new hydrogen molecule plus a hydrogen atom. This is one of the simplest systems to study, as the bimolecular collision contains only three protons and three electrons. r1 denotes the distance between atom 1 and atom 2 and r2 is the distance between atom 2 and atom 3.

The transition state of the reaction between a single hydrogen atom, H, with diatomic hydrogen, H2, must occur when the interatomic distances are equal. This is because the reaction is neither exothermic or endothermic in nature. The transition state must not be late or early, but directly equidistant between reactants and products. As a result, r1 must be equal to r2 and their momenta zero as well. With no oscillation, the gradient, force and acceleration of the particles must also be zero at the transition state.

[[File:Yg8818 int dist hhh.png|left|250 px|thumb|Figure 1ː Interatomic Distances vs Time plot for the H and H2 model]]

When r1 = r2 = 90.775 pm, the above conditions are met and the transition state is observed. This is visible as a single point on the surface plot and as straight lines with no oscillations on the Interatomic Distances vs Time plot, as shown in Figure 1 (left).

[[File:Yg8818 MEP hhh.png|250 px|thumb|Figure 2ː MEP for the H and H2 model]]
[[File:Yg8818 dynamics hhh.png|250 px|thumb|Figure 3ː Dynamics trajectory for the H and H2 model]]

Now that the transition state has been located, the reaction path, which is the Minimum Energy Path (MEP), can be determined. This is a trajectory that corresponds to infinitely slow motion of the particles and can be found by slightly altering the value of one of the interatomic distances. Here, r1 is set equal to 91.775 pm, just 1 pm higher than used for finding the transition state. This trajectory is modelled in Figure 2 (right).

The MEP doesn't account for mass and inertial motion of the atoms. While this provides a useful simplification, the model does not accurately reflect how the particles move during the reaction. This can be resolved by maintaining the current parameters, but using a dynamic calculation type for the interatomic distances vs. time plot. This is displayed in Figure 3 (right). Altering the distance for r2 instead of r1 would yield the same results.

Trajectories with initial conditions in the range r1 = 74 pm, r2 = 200 pm, with -3.1 < p1/ g.mol-1.pm.fs-1 < -1.6 and p2 = -5.1 g.mol-1.pm.fs-1 have shown to be reactive in the above calculations. Here, higher values of momenta have been trialled, while keeping the position values the same, in determining whether the species will still be reactive and the appearance of the trajectory. The results are displayed in Table 1 below.

{| class="wikitable" border=1
! p1/ g.mol-1.pm.fs-1 !! p2/ g.mol-1.pm.fs-1 !! Etot !! Reactive? !! Description of the dynamics !! Illustration of the trajectory
|-
| -2.56 || -5.1 || -414.280 || style="text-align: center;" |Yes || The atoms here approach the transition state with no oscillation at all. Having crossed the saddle point, the resultant products begin to oscillate noticeably. || [[File:Yg8818_tp1.png| 250px]]
|-
| -3.1 || -4.1 || -420.077 || style="text-align: center;" |No || No reaction occurs at these momenta values. It is evident from the contour plot that the molecules approach the transition state while oscillating, but don't have enough kinetic energy to pass over the saddle point. The reactants then return to their original state. || [[File:Yg8818_tp2.png| 250px]]
|-
| -3.1 || -5.1 || -413.977 || style="text-align: center;" |Yes || With an increased amount of momentum for p2, the reactants reach the saddle point without oscillation. The route from transition state to products is full of high energy oscillation (greater than for the first reaction outlined in the table) which can be directly related to the molecules having more kinetic energy. || [[File:Yg8818_tp3.png| 250px]]
|-
| -5.1 || -10.1 || -357.277 || style="text-align: center;" |Yes || After the transition state has been surmounted and products are forming, lots of oscillations can be observed on this contour plot. This is as a result of yet another increase in kinetic energy, although the total energy (which accounts for potential energy as well) is lower than for some of the other examples. || [[File:Yg8818_tp4.png| 250px]]
|-
| -5.1 || -10.6 || -349.477 || style="text-align: center;" |Yes || These conditions display an example of barrier recrossing. Here, transition state is reach by the reactants, but their extremely high kinetic energy results in strong oscillations. These prevent the molecules forming products and instead they revert back to their original state. This occurrence cannot be explained by transition state theory, as ordinarily, once a reaction has passed the saddle point it should progress to form products. || [[File:Yg8818_tp5.png| 250px]]
|}

It can be inferred from Table 1 that an increase in potential (and hence kinetic energy), for either distance, results in molecules with greater oscillation. If the potential energy is too low, transition state won't be reached. If the kinetic energy component is too large, oscillations can interfere with the reaction, resulting in a return to reactants even after the saddle point has been crested. A balance found between the two is essential for reaction to occur.

These calculations are all theoretical and based entirely on transition state theory. Transition state theory complies with several key assumptionsː
# The energy of the particles in the reaction follows the Boltzmann distribution
# The reactants are in constant equilibrium with the transition state structure
# Effects of quantum tunnelling are negligible
# Motion along the reaction coordinate can be separated from other motions and treated classically at the transition state
# Once the transition state has been achieved, the reactants will not reform

This final assumption has shown to be untrue in some circumstances, as in the last example studied (Table 1). Transition state theory does not account for barrier recrossing which is observed frequently in experiment. This could result in transition state theory predicting reactions to have a faster rate than experiment, as a slower rate would be observed with barrier recrossing.

Ignoring effects of quantum tunnelling (as mention above) assumes that the reaction will not take place unless sufficient activation energy is reached to overcome the transition state. However, according to quantum mechanics, particles can still tunnel across the barrier, if the barrier is of finite energy. In experiment, molecules will react sometimes, even if they do not collide with enough energy to scale the transition state barrier. This is an important feature for reactions with small activation energies (tunnelling probability decreases with increasing barrier height) and should not be ignored. Transition state theory will therefore predict that some reactions won't occur, while they may occur successfully in experiment.

In its calculation of reaction rate, transition state theory is limited for reactions occurring at high temperatures. The theory assumes that the system will traverse the lowest energy saddle point on the potential energy surface, which is not always true for systems at high temperatures. At high temperatures, molecules will populate higher energy modes and transitions states will be crosses that are far from the lowest energy saddle point. This can even be observed in this very collision of H with H2 when the experiment is conducted at increased temperatures.

A combination of all of these factors results in transition state theory falling short, in some cases, in its prediction of reaction rate.

== F - H - H System ==

Inspection of the potential energy surfaces for the reactions of F with H - H and H with F - H can be used to classify their energetics; determining if they are exothermic or endothermic reactions. Exothermic reactions have 'early' transition states, as the structure of the transition state closely resembles that of the reactants. Endothermic reactions have 'late' transition states, as the structure of the transition state closely resembles that of the products. This is based on Hammond's Postulate which asserts that if two states (eg. a transition state and an intermediate) occur consecutively in a reaction and have nearly the same energy, their interconversion will only involve a small reorganisation of molecular structure.

The reaction of F with H - H has an early transition state, rendering it an exothermic reaction; the transition state close in position to the reactant bond lengths. Meanwhile, when H collides with F - H, a late transition state is observed indicating an endothermic reaction where the saddle point lies close to the product bond lengths. This can be explained by the bond strength of H - F, which takes a value of 565 KJ mol-1. This is much greater than the bond energy of H - H which is 432 KJ mol-1. More energy is required to break the H - F bond than is released in forming the H - H bond.

The transition state is found when H - F = 181.000 pm and H - H =74.4870 pm, with momenta set to zero for both. The Internuclear Distance vs Time graph (Figure 4) illustrates this well, as a straight line on the plot with no oscillations. This is characteristic of a transition state.

[[File:Yg8818_ts_fhh.png]]
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