ChemWiki - User contributions [en]

Rep:Mod:einstein

2018-05-11T16:59:30Z

Jc2216: /* James Cochrane's Wiki */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> The MEP calculated value (see later) of <math>\Delta\text{H}\simeq</math>-134.016-(-104.004)= -30.012 kcal/mol for the reaction is accurate and correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

The energy of the transition state, <math>\Delta \text{G}^{\ddagger}</math>, in both energy level diagrams (above) is -103.752 kcal/mol. This means that the activation energy of the reaction, <math>E_\text{a}</math>= +0.252 kcal/mol. This very small activation energy is in accordance with the reaction's vigour.

The table below explores the effect of changing the amount of translational energy provided to the system.
{| class="wikitable sortable" border="1"
! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>, <math>p_{AB}=-0.5</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -3 ||[[File:Einsteinfh2blue.png|400px ]] || -96.254 || ✗
|-
| -2 ||[[File:Einsteinfh2blue1.png|400px ]] || -100.754 || ✗
|-
| -1 ||[[File:Einsteinfh2blue2.png|400px ]] || -103.254 || ✗
|-
| 2 ||[[File:Einsteinfh2blue3.png|400px ]] || -98.754 || ✗
|-
| 2.3 ||[[File:Einsteinfh2blue4.png|400px ]] || -97.314 || ✓
|-
| 2.5 ||[[File:Einsteinfh2blue5.png|400px ]] || -96.254 || ✗
|-
| 3 ||[[File:Einsteinfh2blue6.png|400px ]] || -93.254 || ✗
|-
|}
It is clear from the above table that the amount of translational energy provided is not solely responsible for a successful trajectory. Rather, the dominant variable is suggested to be the vibrational phase in HCHB at the moment FA approaches closely.<ref name="e5" /> The following contour diagram proves this point.

[[File:Einsteinweird.png|700px|thumb|none|Low energy Dynamics trajectory with following conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{AB}=-0.8</math> and <math>p_{BC}=0.1</math>. The success of this clearly due to factors other than initial translational energy.]]

Overall, this reaction of is an example of an attractive surface (with an early transition state) - the success of a trajectory on such a surface, according to ''Polanyi's rules'',<ref name="e6" /> is assured by high initial translational energy. However, the results show that other factors such as correct vibrational phase of HCHB are also necessary.
=== H + HF<math>\rightarrow</math> H2 + F===
The reverse reaction, H + HF<math>\rightarrow</math> H2 + F, is therefore endoergic with <math>\Delta\text{H}\simeq</math>-104.004-(-134.016)= +30.012 kcal/mol.

A successful reaction trajectory across this repulsive (late TS) surface is shown below.

[[File:Einsteinfinal.png|700px|thumb|none|Dynamics trajectory with following conditions: <math>r_{BC}=0.94</math>, <math>r_{AB}=3</math>, <math>p_{AB}=-5</math> and <math>p_{BC}=4.3</math>. The TS is encountered late into the reaction.]]

Using similar calculations as used earlier, the approximate location of the late transition state for this reaction was determined to be <math>r_{BC}=0.730</math>, <math>r_{AB}=1.021</math> (see above diagram).

The energy of the reactants was determined using an MEP calculation to give -133.781 kcal/mol (see diagram below).

The energy of the transition state, <math>\Delta \text{G}^{\ddagger}</math> is -74.1589 kcal/mol. This means that the activation energy of the reaction, <math>E_\text{a}</math>= +59.622 kcal/mol.

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>

<ref name= "e6">Z. Zhang, Y. Zhou, D. H. Zhang, G. Czakó and J. M. Bowman, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

File:Einsteinfinal.png

2018-05-11T16:30:40Z

Jc2216:

Rep:Mod:einstein

2018-05-11T15:41:49Z

Jc2216: /* References */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> The MEP calculated value (see later) of <math>\Delta\text{H}\simeq</math>-134.016-(-104.004)= -30.012 kcal/mol for the reaction is accurate and correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

The energy of the transition state, <math>\Delta \text{G}^{\ddagger}</math>, in both energy level diagrams (above) is -103.752 kcal/mol. This means that the activation energy of the reaction, <math>E_\text{a}</math>= +0.252 kcal/mol. This very small activation energy is in accordance with the reaction's vigour.

The table below explores the effect of changing the amount of translational energy provided to the system.
{| class="wikitable sortable" border="1"
! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>, <math>p_{AB}=-0.5</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -3 ||[[File:Einsteinfh2blue.png|400px ]] || -96.254 || ✗
|-
| -2 ||[[File:Einsteinfh2blue1.png|400px ]] || -100.754 || ✗
|-
| -1 ||[[File:Einsteinfh2blue2.png|400px ]] || -103.254 || ✗
|-
| 2 ||[[File:Einsteinfh2blue3.png|400px ]] || -98.754 || ✗
|-
| 2.3 ||[[File:Einsteinfh2blue4.png|400px ]] || -97.314 || ✓
|-
| 2.5 ||[[File:Einsteinfh2blue5.png|400px ]] || -96.254 || ✗
|-
| 3 ||[[File:Einsteinfh2blue6.png|400px ]] || -93.254 || ✗
|-
|}
It is clear from the above table that the amount of translational energy provided is not solely responsible for a successful trajectory. Rather, the dominant variable is suggested to be the vibrational phase in HCHB at the moment FA approaches closely.<ref name="e5" /> The following contour diagram proves this point.

[[File:Einsteinweird.png|700px|thumb|none|Low energy Dynamics trajectory with following conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{AB}=-0.8</math> and <math>p_{BC}=0.1</math>. The success of this clearly due to factors other than initial translational energy.]]

Overall, this reaction of is an example of an attractive surface (with an early transition state) - the success of a trajectory on such a surface, according to ''Polanyi's rules'',<ref name="e6" /> is assured by high initial translational energy. However, the results show that other factors such as correct vibrational phase of HCHB are also necessary.

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>

<ref name= "e6">Z. Zhang, Y. Zhou, D. H. Zhang, G. Czakó and J. M. Bowman, ''J. Phys. Chem. Lett.'', 2012, '''3''', 3416-3419.</ref>
</references>

Rep:Mod:einstein

2018-05-11T15:35:19Z

Jc2216: /* James Cochrane's Wiki */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> The MEP calculated value (see later) of <math>\Delta\text{H}\simeq</math>-134.016-(-104.004)= -30.012 kcal/mol for the reaction is accurate and correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

The energy of the transition state, <math>\Delta \text{G}^{\ddagger}</math>, in both energy level diagrams (above) is -103.752 kcal/mol. This means that the activation energy of the reaction, <math>E_\text{a}</math>= +0.252 kcal/mol. This very small activation energy is in accordance with the reaction's vigour.

The table below explores the effect of changing the amount of translational energy provided to the system.
{| class="wikitable sortable" border="1"
! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>, <math>p_{AB}=-0.5</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -3 ||[[File:Einsteinfh2blue.png|400px ]] || -96.254 || ✗
|-
| -2 ||[[File:Einsteinfh2blue1.png|400px ]] || -100.754 || ✗
|-
| -1 ||[[File:Einsteinfh2blue2.png|400px ]] || -103.254 || ✗
|-
| 2 ||[[File:Einsteinfh2blue3.png|400px ]] || -98.754 || ✗
|-
| 2.3 ||[[File:Einsteinfh2blue4.png|400px ]] || -97.314 || ✓
|-
| 2.5 ||[[File:Einsteinfh2blue5.png|400px ]] || -96.254 || ✗
|-
| 3 ||[[File:Einsteinfh2blue6.png|400px ]] || -93.254 || ✗
|-
|}
It is clear from the above table that the amount of translational energy provided is not solely responsible for a successful trajectory. Rather, the dominant variable is suggested to be the vibrational phase in HCHB at the moment FA approaches closely.<ref name="e5" /> The following contour diagram proves this point.

[[File:Einsteinweird.png|700px|thumb|none|Low energy Dynamics trajectory with following conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{AB}=-0.8</math> and <math>p_{BC}=0.1</math>. The success of this clearly due to factors other than initial translational energy.]]

Overall, this reaction of is an example of an attractive surface (with an early transition state) - the success of a trajectory on such a surface, according to ''Polanyi's rules'',<ref name="e6" /> is assured by high initial translational energy. However, the results show that other factors such as correct vibrational phase of HCHB are also necessary.

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>
</references>

File:Einsteinweird.png

2018-05-11T15:18:52Z

Jc2216:

Rep:Mod:einstein

2018-05-11T15:15:12Z

2018-05-11T14:44:53Z

Jc2216: /* F + H2 \rightarrow H + HF */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> The MEP calculated value (see later) of <math>\Delta\text{H}\simeq</math>-134.016-(-104.004)= -30.012 kcal/mol for the reaction is accurate and correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

The energy of the transition state, <math>\Delta \text{G}^{\ddagger}</math>, in both energy level diagrams (above) is -103.752 kcal/mol. This means that the activation energy of the reaction, <math>E_\text{a}</math>= +0.252 kcal/mol. This very small activation energy is in accordance with the reaction's vigour.

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>
</references>

Rep:Mod:einstein

2018-05-11T14:39:07Z

Jc2216: /* James Cochrane's Wiki */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> The MEP calculated value (see later) of <math>\Delta\text{H}\simeq</math>-134.016-(-104.004)= -30.012 kcal/mol for the reaction is accurate and correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

The energy of the transition state in both energy level diagrams (above) is -103.752 kcal/mol. This means that the activation energy of the reaction, <math>\Delta E_a</math>= +0.252 kcal/mol. This very small activation energy is in accordance with the reaction's vigour.

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>
</references>

Rep:Mod:einstein

2018-05-11T14:24:30Z

Jc2216: /* F + H2 \rightarrow H + HF */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> Although the calculated value of <math>\Delta\text{H}\simeq</math>-20 kcal/mol (for the trajectory below) is not very accurate, it correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The energy of the products was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.81</math>) to give -134.016 kcal/mol (see diagram below).

[[File:Einsteinfh2product.png|500px|thumb|none|(MEP) energy level diagram going from TS to products. ]]

The energy of the reactants was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}=1.82</math>) to give -104.004 kcal/mol (see diagram below).

[[File:Einsteinfh2product1.png|500px|thumb|none|(MEP) energy level diagram going from TS to reactants.]]

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>
</references>

File:Einsteinfh2product1.png

2018-05-11T14:23:11Z

Jc2216:

File:Einsteinfh2product.png

2018-05-11T14:07:54Z

Jc2216:

Rep:Mod:einstein

2018-05-11T14:04:12Z

Jc2216: /* James Cochrane's Wiki */

==James Cochrane's Wiki==
===H + H2 <math>\rightarrow</math> H + H2===
In the following analysis of the collision between an H atom and an H2 molecule, the atoms are constrained to be collinear (collision angle is 180). It is important to realize that although the potential barrier is least for collinear attack, other lines of attack are feasible and contribute to the overall rate of reaction in reality. Additionally, effects of quantum mechanical tunnelling on reactivity are ignored; we just consider the classical trajectories of particles over surfaces.<ref name="e1" />

Two parameters are required to define the nuclear separation: the HA-HB separation <math>r_{AB}</math> and the HB-HC separation <math>r_{BC}</math>. A plot of the total energy of the system against <math>r_{AB}</math> and <math>r_{BC}</math> gives the potential energy surface of this reaction.

[[File:einsteinSurface1new.png|700px|thumb|none|Generic potential energy surface for the H + H2 <math>\rightarrow</math> H + H2 reaction.]]

Although the 3D surface provides a useful visualization tool, contour diagrams (with contours of equal potential energy) will be more suitable for analysis.

[[File:einsteinContour1new.png|700px|thumb|none|Generic contour diagram for the H + H2 <math>\rightarrow</math> H + H2 reaction. Equilibrium bond lengths are shown (strictly they represent the relevant bond lengths when the third atom is infinitely far away).]]

Here, I use the notation <math>x=r_{AB}</math>, <math>y=r_{BC}</math> and <math>V_{x}=\frac{\partial V}{\partial x}</math>, where <math>V</math> is the potential energy function.

Suppose that <math>V(x,y)</math> has a local extremum at a point <math>(x_0,y_0)</math>, then <math>V_{x}(x_0,y_0)=V_{y}(x_0,y_0)=0</math>. These properties hold true at both a transition state and at minimum. To distinguish between these two structures we must use the following formula:
<math>D \equiv V_{xx}V_{yy} - V_{xy}V_{yx}</math>. If <math>D>0</math> and <math>V_{xx}>0</math>, we are at a minimum. Whereas, if <math>D>0</math> and <math>V_{xx}<0</math>, we are at a transition state. Additionally, if <math>D<0</math>, we are at a saddle point.<ref name="e2" />

====Question 2====
[[File:Einstein1.png|700px|thumb|none|Successful reactive trajectory for the reaction of HA + HBHC <math>\rightarrow</math> HC + HAHB. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>. The approximate location of the transition state (TS) is shown.]]

The transition state is located near <math>r_{AB}=r_{BC}=0.91</math>. To improve upon this value we test a range of values <math>r_{AB}=r_{BC}</math> and <math>p_{AB}=p_{BC}=0</math>: akin to seeing which side of the hill a stationary ball rolls down.

{| class="wikitable sortable" border="1"
! Trial !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Internuclear Distances vs Time!! class="unsortable" | Successful?
|-
| 1 || 0.9 || 0.9 || 0 || 0 ||[[File:Einsteintrail1.png|300px]] || ✗
|-
| 2 || 0.91 || 0.91 || 0 || 0 ||[[File:Einsteintrail2.png|300px]] || ✗
|-
| 3 || 0.92 || 0.92 || 0 || 0 ||[[File:Einsteintrail3.png|300px]] || ✓
|-
| 4 || 0.917 || 0.917 || 0 || 0 ||[[File:Einsteintrail4.png|300px]] || ✓
|-
|}

After 4 tested values (with 500 steps), the transition state occurs when the system has roughly the arrangement: <math>r_{AB}=r_{BC}=0.917</math>. Specifically, the transition state is a set of configurations (a line on the potential energy surface which passes through a saddle point) associated with a critical geometry such that every trajectory that goes through this geometry reacts.

{| class="wikitable sortable" border="1"
! Calculation Type !! <math>r_{AB}</math> !! <math>r_{BC}</math> !! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram
|-
| MEP || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep1.png|300px|thumb|none| This trajectory (towards the products) is of least energy along the valley floor because each iteration sets the velocity (and kinetic energy) to zero - preventing any movement up the valley walls.]]
|-
| Dynamics || 0.917 || 0.927 || 0 || 0 ||[[File:Einsteinmep2.png|300px|thumb|none| Whereas this trajectory (towards the products) does not take the route of least potential energy because the non-zero velocity of the reactants allows the valley walls to be traversed.]]
|-
| MEP || 0.927|| 0.917 || 0 || 0 ||[[File:Einsteinmep4.png|300px|thumb|none| When the initial conditions are moved onto the reactant "side" of the transition state the trajectory goes to reactants. It has the same properties as the other MEP calculation above. ]]
|-
| Dynamics || 0.927 || 0.917 || 0 || 0 ||[[File:Einsteinmep3.png|300px|thumb|none| Upon a change in conditions, the arrangement falls to the reactants. It has the same properties as the other Dynamics calculation above. ]]
|-
|}
In an attempt to run the minimal kinetic energy reverse reaction, we take the output average radii and momenta from the above calculations. It is expected that the reaction trajectory will almost get to the transition state and then return via the reactant channel. Here is the result of such a collision.

[[File:Einsteinreverse1.png|600px|thumb|none|The reverse reaction approaches from the "product" channel with no vibration energy, almost reaches the transition state (TS) and then returns via the "product" channel in a vibrationally excited state. Therefore, HAHB had insufficient kinetic energy. Conditions: <math>r_{BC}=9.06</math>, <math>r_{AB}=0.74</math>, <math>p_{BC}=-2.46</math> and <math>p_{AB}=-1.19</math>.]]

The following table shows the effect of higher collision momenta on the success of a reaction with fixed radii.

{| class="wikitable sortable" border="1"
! <math>p_{AB}</math> !! <math>p_{BC}</math> !!class="unsortable" | Contour Diagram (<math>r_{BC}=0.74</math>, <math>r_{AB}=2.0</math>) !! Total energy !! class="unsortable" |Reactive?
|-
| -2.5 || -1.25 ||[[File:Einsteinq1.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -99.02 || ✓
|-
| -2 || -1.5 ||[[File:Einsteinq2.png|400px|thumb|none| HA approaches a vibrating HBHC molecule, collides unsuccessfully; a vibrationally excited HBHC molecule re-emerges. The phase of vibration is insufficient for the reaction. ]] || -100.46 || ✗
|-
| -2.5 || -1.5 ||[[File:Einsteinq3.png|400px|thumb|none| HA approaches a slowly vibrating HBHC molecule, reacts successfully; a vibrationally excited HAHB molecule emerges. ]] || -98.96 || ✓
|-
| -5.0 || -2.5 ||[[File:Einsteinq4.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state but recrosses this region; a vibrationally excited HBHC molecule re-emerges. ]] || -84.96 || ✗
|-
| -5.2 || -2.5 ||[[File:Einsteinq5.png|400px|thumb|none| HA approaches a non-vibrating HBHC molecule, passes the transition state and recrosses this region several times; a highly vibrationally excited HAHB molecule emerges. ]] || -83.42 || ✓
|-
|}

The main assumption of transition state theory (TST) is that there is a crucial configuration of no return called the transition state. If molecules pass through this spatial configuration then it is inevitable that they will form products from this encounter.<ref name="e1" /> In strict terms, this "inevitability" is only certain once the nascent products are significantly separated from each other.<ref name="e3" />

The experimental result with a calculated energy of -84.96 (see above table) does not agree with TST predictions because this trajectory crosses through the transition state several times but does not afford products.
===F + H2 <math>\rightarrow</math> H + HF===
It is known that the F + H2 <math>\rightarrow</math> H + HF reaction is highly exoergic (exothermic) with a value of <math>\Delta\text{H}\simeq</math>-32 kcal/mol.<ref name="e4" /> Although the calculated value of <math>\Delta\text{H}\simeq</math>-20 kcal/mol (for the trajectory below) is not very accurate, it correctly implies that the H-F bond is much stronger than the H-H bond.

[[File:Einsteinfh2.png|700px|thumb|none|Successful reactive trajectory for the reaction of HCHB + FA <math>\rightarrow</math> HC + HBFA. Conditions: <math>r_{BC}=0.74</math>, <math>r_{AB}=2.3</math>, <math>p_{BC}=0</math> and <math>p_{AB}=-2.7</math>.]]

For the successful trajectory above, energy is channelled efficiently into high states of HBFA vibration (shown by the oscillating AB distance). This high vibrational excitation in product HBFA arises from the release of HC-HB bond repulsion while the HB-FA distance is still large; the large zero point vibration in HCHB introduces variability in the observed HBFA vibrational distribution.<ref name="e5" /> In simpler terms, trajectories that give rise to large vibrational energy of product(s) typically cut the corner of the energy surface and approach the exit valley from the side.

One way of confirming the formation of vibrationally excited HBFA is the method of ''infrared chemiluminescence'' in which emission of infrared radiation can be detected as HBFA returns to its ground state. The populations of the vibrational states of the product can then be determined. An alternative method relies on ''laser-induced flourescence'' in which HBFA is excited from a specific vibration-rotation level using lasers. ''Crossed molecular beams'' can also be employed.<ref name="e1" />

By running 20 Dynamics calculations (1000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>1.75<r_{AB}<1.85</math>), the approximate location of the early transition state for this reaction was determined to be <math>r_{BC}=0.745</math>, <math>r_{AB}=1.8107</math> (see above diagram). The early transition state shows that this reaction surface is ''attractive'' and it is well known that such surfaces proceed more efficiently if the energy supplied is largely translational.<ref name="e1" />

The '''energy of the products''' was determined using an MEP calculation (400,000 steps, <math>p_{BC}=p_{AB}=0</math>, <math>r_{BC}=0.745</math> and <math>r_{AB}<1.81</math>) to give -134.016 kcal/mol (see diagram below).

==References==
<references>

<ref name= "e1">P. Atkins and J. de Paula, ''Atkins' Physical Chemistry'', Oxford University Press, Oxford, 10th edn., 2014, ch. 21, 881-928. </ref>

<ref name= "e2"> Wolfram Mathworld, http://mathworld.wolfram.com/SecondDerivativeTest.html, (accessed May 2018).</ref>

<ref name= "e3"> R. D Levine, ''Molecular Reaction Dynamics'', Cambridge University Press, Cambridge, 2009, 202-212. </ref>

<ref name= "e4"> J. Chen, Z. Sun and D. H. Zhang, ''J. Chem. Phys.'', 2015, '''142''', 1-11.</ref>

<ref name= "e5">J. C. Polanyi and J. L. Schreiber, ''Faraday Discuss. Chem. Soc.'', 1977, '''62''', 267-290.</ref>
</references>

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