MRD:solomona

2018-05-18T16:16:04Z

Sa7516: /* Activation Energy */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG|thumb|center|MEP Plot]]
[[File:Sa7516-1DYN plot.PNG|thumb|center|Dynamics Plot]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

As before, the momentum time graph is shown bellow as evidence that the transition state was found.

[[File:Solo Q2.PNG]]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! rAB !! rBC !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions were rAB= 1.80 and rBC= 0.74 Å, no momentum.

[[File:SaHF energy 2.png]]

[[File:SaHF energy (step)2.png]]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions were rAB= 1.8195 Å and rBC= 0.7545 Å, no momentum.

[[File:SaHF energy 1.png]]

[[File:SaHF energy 1 (step).png]]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[[File:Sol-ENERGY DIST OF HF.png]]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

MRD:solomona

2018-05-18T16:14:43Z

Sa7516: /* The Mechanism of Reaction Energy Release */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG|thumb|center|MEP Plot]]
[[File:Sa7516-1DYN plot.PNG|thumb|center|Dynamics Plot]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

As before, the momentum time graph is shown bellow as evidence that the transition state was found.

[[File:Solo Q2.PNG]]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! r1 !! r2 !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions were rAB= 1.80 and rBC= 0.74 Å, no momentum.

[[File:SaHF energy 2.png]]

[[File:SaHF energy (step)2.png]]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions were rAB= 1.8195 Å and rBC= 0.7545 Å, no momentum.

[[File:SaHF energy 1.png]]

[[File:SaHF energy 1 (step).png]]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[[File:Sol-ENERGY DIST OF HF.png]]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

File:Sol-ENERGY DIST OF HF.png

2018-05-18T16:14:16Z

Sa7516:

MRD:solomona

2018-05-18T16:13:00Z

Sa7516: /* Activation Energy */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG|thumb|center|MEP Plot]]
[[File:Sa7516-1DYN plot.PNG|thumb|center|Dynamics Plot]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

As before, the momentum time graph is shown bellow as evidence that the transition state was found.

[[File:Solo Q2.PNG]]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! r1 !! r2 !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions were rAB= 1.80 and rBC= 0.74 Å, no momentum.

[[File:SaHF energy 2.png]]

[[File:SaHF energy (step)2.png]]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions were rAB= 1.8195 Å and rBC= 0.7545 Å, no momentum.

[[File:SaHF energy 1.png]]

[[File:SaHF energy 1 (step).png]]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[FIGURE]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

MRD:solomona

2018-05-18T16:10:15Z

Sa7516: /* Activation Energy */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG|thumb|center|MEP Plot]]
[[File:Sa7516-1DYN plot.PNG|thumb|center|Dynamics Plot]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

As before, the momentum time graph is shown bellow as evidence that the transition state was found.

[[File:Solo Q2.PNG]]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! r1 !! r2 !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions: rAB= 1.80 Å rBC= 0.74 Å

[[File:SaHF energy 2.png]]

[[File:SaHF energy (step)2.png]]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions: rAB= 1.8195 Å rBC= 0.7545 Å

[[File:SaHF energy 1.png]]

[[File:SaHF energy 1 (step).png]]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[FIGURE]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

MRD:solomona

2018-05-18T16:09:10Z

MRD:solomona

2018-05-18T16:00:18Z

Sa7516: /* A Comparison of MEP and Dynamics Calculations */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG|thumb|center|MEP Plot]]
[[File:Sa7516-1DYN plot.PNG|thumb|center|Dynamics Plot]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

[PLOT GOES HERE]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! r1 !! r2 !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions: rAB= 1.80 rBC= 0.74

[ENERGY SURFACE]

[ENERGY PLOT]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions: rAB= 1.8195 rBC= 0.7545

[ENERGY SURFACE]

[ENERGY PLOT]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[FIGURE]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

MRD:solomona

2018-05-18T15:54:27Z

Sa7516: /* H+H2 System */

===H+H2 System===

====Gradients at Transition States and Minima====

At minima and transition states (i.e. saddle points) the derivative of the potential energy will be zero; ∂V(ri)/∂ri = 0. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative is positive, so ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification. We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{r1r1} f_{r2r2} - {f_{r1r2}}^2 </MATH>

If D > 0 and fr1r1 < 0 then we have a local minimum.

If D > 0 and fr1r1 > 0 then we have a local maximum.

If D < 0 then we have a saddle point.

For D=0, higher order tests must be used.

====Locating the Transition State from an “Internuclear Distances vs Time” plot====

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for rAB and rBC. Using simple trail and error with rAB = rBC and pAB = pBC=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distances against time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

====A Comparison of MEP and Dynamics Calculations====

Using the values rAB = 0.9075 and rBC = 0.9175 with pAB = pBC=0, the behavior of the system as it moves away from the transition state was analysed with both mep (minimum reaction path) and dynamics calculations.

[[File:Sa7516-1MEP.PNG]]
[[File:Sa7516-1DYN plot.PNG]]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

====Reactive and Unreactive Trajectories====

Analysis of changes in momentum as the system moves away from the transition state shows that the final momentum values are 0.8<pAB<1.5 pBC=-2.5. This shows that a system with -1.5<pAB<-0.8 and pBC=-2.5 will lead to reaction.

[[File:SAFigure 2.png]]

It seems fair to assume that any system with initial momenta higher than these would also lead to reaction. This claim was tested by investigating a range of momenta values, using initial positions rAB = 0.74 and rBC = 2.0.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react? !! Trajectory Plot !! Description of Trajectory
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|200px]] || The distance r2 steadily decreases but upon collision there is insufficient energy for reaction. The initial Ha-Hb bond is reformed and the r2 goes back up.
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|200px]] || The distance r2 steadily decreases. Upon collision, the system is able to overcome the energy barrier, passing through the transition state and producing the products. After this r1 steadily increases.
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|200px]] || The products from, but due to excess kinetic energy the system recrosses the energy barrier. It passes once more through the transition state and the reactants are reformed.
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|200px]] || The products form, recross the barrier back to the reactants side, then recross once more to finally form the products.
|}

We find that the hypothesis proves false, as some simulations using momenta values higher than the predicted requirements (e.g. the system with pAB=-2.5 and pBC=-5) failed to react.

====The Assumptions of Transition State Theory====

The two most basic assumptions of transition state theory are:
* The separation of nuclear and electronic motion (known as the Born-Oppenheimer approximation in quantum mechanics)

* The assumption that molecules are distributed among their energy states in accordance to the Maxwell-Boltzmann distribution.

These assumptions are common to many other theories. Transition state theory also has three more assumptions, unique to it:
# Molecular systems that have crossed the transition state to form products cannot reform reactants
# In the transition state, motion along the reaction coordinate may be separated from other motions and be treated as a classical translation.
# Even in the absence of an equilibrium between products and reactants, systems at the transition state becoming products are distributed among their states following the Maxwell-Boltzmann laws.
It should be noted that the third assumption is not formally necessary, as it follows from the first. [REFERENCE CKD]

The results form the last two trials shown in the table above directly contradict the first assumption of Transition State Theory, as the system can clearly be seen crossing the transition state multiple times. In the cases tested, Transition State Theory proves to be an oversimplification.

===F-H-H System===

====Reaction Energetics====

The enthalpy of the H-H bond is ca. 432 kJ mol-1 , and for the H-F bond it is ca. 565 kJ mol1. Following from this, it can be seen that the F + H2 → HF + F reaction is exothermic, as the energy released by the formation of the H-F bond is greater than the energy required to break the H-H bond. Overall, ca. 133 kJ mol-1 can be expected to be released. Conversely, the opposite is true for the H + HF → H2 + F; ca 133 kJ mol-1 will be taken in by the system. [REFERENCE atkins]

====The Position of the Transition State====

Using the same method outlined above for the H-H-H system, the approximate position of the transition state was estimated. The following values were found:

F-H distance: ca. 1.8095 Å

H-H distance: ca. 0.7445 Å

[PLOT GOES HERE]

==== Activation Energy ====
Total energy at the transition state was compared to the total energy at each of the respective starting points. Literature values for the lengths of each bond informed the initial conditions of the starting points. The relevant total energy values are shown below:

{| class="wikitable" border=1
! System !! r1 !! r2 !! Total Energy
|-
| F-H-H (Transition State) || 1.8095 || 0.7445 || -103.752
|-
| F-H H || 0.91 || 100 || -133.955
|-
| F H-H || 100 || 0.74 || -104.020
|}

From these values, the following activation energy values were calculated:

'''H + HF → H2 + F'''

Activation Energy = 30.203 kcal/mol

'''F + H2 → HF + F'''

Activation Energy = 0.268 kcal/mol

Analysis of an mep plot of energy over time was also used to calculate the activation energy of the reactions. Potential energy at appropriate starting positions (neighboring the transition state, slightly to the side of the desired starting materials) and the respective final positions were compared to find activation energy values.

'''H + HF → H2 + F'''

Starting conditions: rAB= 1.80 rBC= 0.74

[ENERGY SURFACE]

[ENERGY PLOT]

Activation Energy = 30.26718247 kcal/mol

'''F + H2 → HF + F'''

Starting conditions: rAB= 1.8195 rBC= 0.7545

[ENERGY SURFACE]

[ENERGY PLOT]

Activation Energy = 0.235073551 kcal/mol

====The Mechanism of Reaction Energy Release====

Following a reaction, excess potential energy is converted into kinetic energy in the translational and vibrational modes of the product molecules. The specific distribution across these modes is dependent on the potential energy surface of the reaction. Surfaces with "late" or "repulsive" barriers, which occur in the exit channel where the products are separating, will have energy transferred predominantly into product translational modes, while "early" or "attractive" barriers (occurs in the entrance channel while reactants are approaching" will have generally favor vibrational excitement. The reasoning behind this can be seen if one considers the changes in bonding that occour during each respective type of barrier. Consider an A+BC system. For a late barrier the energy is released after the A-B bond has already formed and the B-C distance is changing; this motion corresponds to product separation so energy is transferred mainly into relative translation. An early barrier occurs while A-B is changing, so energy is transferred into this bond leading to vibrations [CDK]. From Hammonds postulate (i.e. the transition state resembles the species nearest to it in energy) we known that the F + H2 reaction has an early transition state, and so the H-F product is highly vibrationally excited. This can be seen from the changes in momentum shown the figure below.

[FIGURE]

The distribution of energy across the modes may be measured by taking measurements of pressure and electromagnetic emissions. As all the species present are gaseous and the number of particles remains constant over the reaction, increases in transnational lead to more frequent and more energetic collisions of the gas with the container. Increases in translational energy lead to increases in pressure. Vibrational excitement will result in emission of infra-red radiation, following internal conversion upon the relaxation of excited molecules to their ground state. It must be noted that temperature cannot be used, as population of both modes lead to increased temperature. [H5]

Alternative methods of examining the product's energy state distribution are also possible. Most notably one could examine the nature of any electromagnetic emissions, looking for Doppler broadening. Doppler broadened peaks arise from high translational excitation. As total energy is always conserved, narrow peaks (low translational excitation) implies high vibrational excitation. [H6]

====Distribution of Energy and the Efficiency of Reaction====

The location of energy barrier also plays a key role in the formation of products, as it controls which distribution of energy modes in the reactants is most likely to result in reaction. For a given transition state, the favored reactant energy distribution is the inverse of the favored product energy distribution. Vibrational energy is most effective for passage over late barriers - high translational energy may not lead to reaction even if provided in exccess of the barrier height. An excess of translational energy may simply lead to mollecules "coliding" with the replusive walls of the potential surface, and "rolling" back down into the reactant well. Conversely, an early barrier is best overcome by translational energy. Excess vibrational energy will lead to molecules oscillating from side to side yet never reaching the top of the barrier. [CDK] These rules for understanding simple reaction dynamics are known as the Polanyi's rules. As F + H2 has an early transition state, high translatoinal energy is required for efficient reaction. The reverse is true for H + HF.

EFFECTIVE REACTION -
2.3
0.74
-2
-1.5
[FIGURE]

high momentum

File:Sa7516-1DYN plot.PNG

2018-05-18T15:54:07Z

Sa7516:

MRD:solomona

2018-05-16T13:09:07Z

Sa7516:

What value do the different components of the gradient of the potential energy surface have at a minimum and at a transition structure? Briefly explain how minima and transition structures can be distinguished using the curvature of the potential energy surface.

At minima and transition states (i.e. saddle points) the potential energy, given by ∂V(ri)/∂ri=0, will be zero. We differentiate between transition states and minima by taking further derivatives. For minima the second derivative, ∂2V(ri)/∂ri2>0. A negative value (<0) will be given by maximum points. If the second derivative equates to 0, the point may be a minimum, maximum or saddle point, and further testing will be required for identification.

We must calculate the second derivative test discriminant, D, given by:

<MATH> D = f_{xx} f_{yy} - {f_{xy}}^2 </MATH>

If D > 0 and f_{xx} > 0 then we have a local minimum.
If D > 0 and f_{xx} < 0 then we have a local maximum.
If D < 0 then we have a saddle point.
For D=0, higher order tests must be used.

[include figure]

At the transition state, the three H atoms do not oscillate relative to each other. This fact can be used to locate the position of the transition state, as computation analysis will show constant values for r1 and r2. Using simple trail and error with r1 = r2 and p1 = p2=0, an estimated inter-atomic distance of 0.9075 Å was found for the transition state. A plot of internuclear distaneces vs time can be found below, providing evidence that very little oscillation occurs at these values.

[[File:Solo_Q2.PNG]]

[MEP plot]
[Dynamics plot]

The minimum energy path (mep) follows the valley floor. At each time step in the calculation velocities are set to 0, so this trajectory corresponds to infinitely slow motion. While useful for characterizing reactions, it fails to provide an accurate account of the motion of atoms. The dynamics plot shows the oscillation of the H atoms.

{| class="wikitable" border=1
! p1 !! p2 !! Total Energy (kJ mol-1) !! Does it react?
|-
| -1.25 || -2.5 || -99.018 || YES || [[File: saSP-1.png|250px]]
|-
| -1.5 || -2.0 || -100.456 || NO || [[File: saSP-2.png|250px]]
|-
| -1.5 || -2.5 || -98.956 || YES || [[File: saSP-3.png|250px]]
|-
| -2.5 || -5.0 || -84.956 || NO || [[File: saSP-4.png|250px]]
|-
| -2.5 || -5.2 || -83.416 || YES || [[File: saSP-5.png|250px]]
|}

[ADD headers, methodology. mention what REACTION starting conditions and what calculation type you used. concluding comment below all the trajectories of what you learned from this table]