### Reaction Rate Constants from First Principles

In this computational experiment, the rate constant for the prototypical hydrogen exchange reaction

H + H2 ---> H2 + H (1)

will be calculated. A potential energy surface (PES) for a simple one-dimensional reaction coordinate will be generated, and subsequently plotted via a contouring program. From the PES, the activation energy Ea can be calculated; this, combined with the Arrhenius preexponential factor A, will give the reaction rate constant as follows:

k = Aexp(-Ea/RT) (2)

In addition, we will gain valuable information on the structure of the transition state from study of the PES for reaction (1).

### Procedure

We will use Spartan Student v3.0.2 to calculate the single-point energies for reaction (1) at a variety of H-H-H distances. Our labeling convention will be as follows:

We will fix r13 and vary r23 over the range 0.75A - 1.55A by 0.1A increments (A=Angstrom), calculating the energy using the Hartee-Fock approximation. We then change r13 and repeat the calculation, varying r23.

To begin making calculations, start Spartan Student V3.0.2 from the Start button/All Programs menu on any computer in Sc 270, 271, or 267.

From the main Spartan screen, select File/New. A Builder window appears with a palette of molecular model parts on the right side of the screen. Click on the Inorganic tab, and click on H, and click on the tool that looks like -*- Then click in the green area of the screen - a gray H atom with two yellow valences appears. In the Inorganic tab, click on H, click on the tool that looks like -* , and click on one of the yellow valences. Another H atom will appear. Click on the other valence and another H atom will appear.

In the toolbar at the top of the screen click on the button with the large V (you are now in view mode.)

Pull down the Geometry menu and select Measure distance. Select H1 (one of the end H atoms) and H3 (the middle H) by clicking on them (they'll be highlighted.) In the lower right of the screen you will see Distance(H1,H3)= (if you don't see this, use the Maximize icon in the far upper right of the Windows screen.) Click in the distance Distance (H1,H3) dialog box and set the distance to 0.75A. Hit enter. Select the other end H atom (H2) and the middle H atom (H3), and in the Distance(H2,H3) dialog, set this distance also to 0.75A. Hit enter.

Pull down the Setup menu and select Calculations. On the Calculate pull down, select Single Point Energy with Hartee-Fock/6-31G* (you are choosing the computational model and a set of Gaussian functions with which to represent the atomic orbitals.) Leave the Total Charge set to neutral, and set the Multiplicity to Doublet. Click Submit, and the run will start. You will be asked to name the run; do so, and click Save. You will be told that the run has started (click OK) and that the run has finished (click OK.)

To record the energy of the run (r13=0.75A, r23=0.75A), pull down the Display menu and select Output. Scroll down the output file until you see E(HF) = . Record this energy, along with both H-H distances in your lab notebook. (For comparison purposes, if you've done everything properly, you should get E(HF) = -1.5687831 au for both H-H distances at 0.75A).

Pull down the Geometry menu, and make sure Measure distance is selected. Select H2 and H3, and set the distance to 0.85A (leave H1-H3 alone.) Select Calculations from the setup menu and make sure that all parameters are as given above, submit the job, display the output, and record the H-H distances and the energy.

Repeat with H1-H3=0.75A, varying H2-H3 from 0.85A to 1.55A by 0.1A increments. Record all distances and energies in your lab notebook.

Next, change H1-H3 to 0.80A, and repeat the calculations, varying H2-H3 from 0.75-1.55A by 0.1A.

Do this until you have changed H1-H3 all the way out to 1.55A. Record all data in your lab notebook.

Pay attention to the energies as r23 varies for a given r13 - the attentive student will be able to save a large amount of computational time by noticing that the energies are symmetric!

When you have completed your runs, use Notepad or excel to create an ASCII file called 'surface.dat'. Your file should have three columns separated by tabs. The first column will contain values of r12, the second column the values of r23, and the third column the total energy corresponding to these values. Save this file to your f: drive or a flashdrive.

We will be using a plotting/data analysis program called Origin for Windows (MicroCal, Inc 1992) to display the PES data for reaction 1. We will discuss the use of this program in laboratory lecture. Import your ASCII file ('surface.dat') into the Origin 3D + Contour module. Select column C as the z-axis, and put the data onto a grid. Make a contour plot of your data, and experiment with three-dimensional plotting as a means to display your data.

From your numerical data and your contour plot, identify the minimum energy path for reaction (1) to occur. Make a 2-dimensional plot of energy versus the reaction coordinate [r12-r23]. Calculate the activation energy in kJ mol-1 , and compare your results to accepted values. Propose an explanation if your value differs.

Comment on the geometry of the transition state. What are the bond lengths in the transition state? Are they symmetric? Are they longer than the equilibrium H-H bond lengths (based upon your earlier calculation)? Given that the preexponential factor for reaction (1) is about 109 s-1 at 300 K, find the rate constant for reaction (1) at this temperature.

### The Report

Follow the usual guidelines for laboratory reports. Include a contour plot calculated from your data, a three-dimensional plot of the reaction surface, and a plot of energy versus reaction coordinate for the lowest-energy pathway. Be sure to address the questions listed above.

### References

Adamson, A. "A Textbook of Physical Chemistry", 3rd Ed., Academic Press, New York 1986 and references therein.

Hanna, M.W. "Quantum Mechanics in Chemistry", 3rd Ed., Benjamin-Cummings, Menlo Park, CA 1981.

Huber, K.P. and Herzberg, G. "Constants of Diatomic Molecules", Van Nostrand, New York 1979.

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