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Potential Energy Functions for Hydrogen Halides

Leslie Schwartz

This group of exercises will show students how atoms and molecules interact.

In Exercise 1, students will construct potential energy functions for a series of diatomic molecules by varying the distance between the two atoms in the molecule and calculating the energy of the system at each distance. From these curves, students can calculate and compare the equilibrium bond lengths and well depths (approximate dissociation energies) of the molecules.

Because potential energy functions are so important in explaining basic chemical phenomena (and because the calculations are so fast), Exercise 1 is appropriate for a freshman chemistry class. The instructor could take the opportunity to explain that these potential energy functions are not intuitive (why should the nucleus-electron attractions predominate at longer distances than the nucleus-nucleus and electron-electron repulsions?) and that to calculate them, the machinery of quantum mechanics is required. Morse functions could then be introduced as a convenient set of mathematical functions which approximate the potential energy functions.

In Exercise 2 and Exercise 3, students will construct three-dimensional potential energy surfaces for a series of chemical reactions consisting of head-on collisions between a neutral atom and a diatomic molecule. They will be asked to identify important regions on the surfaces (reactant, product, transition state, reaction coordinate, etc.) and will calculate the energy and bond lengths of the transition state, and the activation energies and the DH value, for each reaction.

Exercise 2 and Exercise 3 cover material that is normally presented in a physical chemistry class. The hardest part of this material is the visualization of the surfaces, and the beauty of having students construct and work with their own surfaces is that they can spend the necessary time rotating their surfaces until they find the orientation that "clicks" for them.

The instructions for the three exercises are given assuming that CAChe molecular modeling software is being used; the exercises can be performed using other software as well.

Exercise 1.

In this exercise you will construct potential energy functions for the following series of diatomic molecules: H-H, H-F, H-Cl, H-Br, and H-I. For each molecule:
  1. Build the molecule using the CAChe editor. Choose SELECT TOOL from the TOOL menu and select one of the atoms in the molecule. Hold down the shift key and select the other atom. Select ATOM DISTANCE from the ADJUST menu. Click on "Define Geometry Label and" and then on "Search" in the bond length window. Search between 0.3 and 7.0 A using 70 steps. Click on "Apply" and then on "Done" to close the bond length window. Save the molecule file you have created in order to save the bond length search label you just created.
  2. Select MOPAC from the APPLICATIONS menu. Choose the "Rigid Search" calculation type. Choose the "Reaction Coordinate" search type and then click on "Done." Type in "Geo-ok" under "Extra Keywords" under "Details." (Geo-ok tells the program not to worry about the fact that for some of the calculations, your atoms are so close together that the situation is physically unreasonable.) Run MOPAC.
  3. Choose VISUALIZER from the APPLICATIONS menu. Open the filename you previously created which now has a .Map extension. You will see two windows. The left window will contain a potential energy curve similar to that shown in Figure 1. The right window contains the molecule with the value of its current bond length displayed on the molecule and the energy of the molecule (which depends on the bond length) displayed at the top of the window.

    Figure 1. HF.Map showing the dependence of potential energy on bond length in H-F.
  4. Click in the left window and move the dot along the potential energy curve by clicking along the curve with the mouse or by using the left and right arrow keys. Watch both windows and note how the potential energy of the molecule changes as the distance between the two atoms changes.
  5. Record the bond length and energy at the minimum of the potential curve, and the energy of the dissociated molecule (i.e. the energy of the separated atoms).

Make a table of equilibrium bond lengths, well depths (De values), and energies of the dissociated molecules for the five molecules studied and discuss/explain the trends. Compare your calculated bond lengths and well depths to literature values. [Note: The actual ground state energy of the molecule is De minus the zero point vibrational energy. The zero point energies are neglected in Exercise 2 and Exercise 3 below.]

Approximate results:

HI HCl HF
Bond Length(A) 1.60 1.26 0.83
De(kcal/mol) 195 205 259
E(atoms)(kcal/mol) 187 181 184

Question: How would the potential functions that you calculate differ if you use molecular mechanics instead of MOPAC? Explain.

Exercise 2.

In this exercise you will construct a potential energy surface for the reaction:

F(g) + H-F(g) = F-H(g) + F(g)(1)

You will study the simplest case, where the collision is head-on (i.e. it occurs with a constant 180 degree F-H-F bond angle)

  1. Build the fictitious molecule, F-H-F, using the CAChe editor. Choose the SELECT TOOL from the TOOL menu and select one of the F atoms. Hold down the shift key and select the H atom. Select ATOM DISTANCE from the ADJUST menu. Choose "Define Geometry Label and" and then "Search" from the bond length window. As a first start, search from 0.5 A before the energy minimum to 4.0 A, using 40 steps. [Once you see the results and become familiar with these kind of surfaces, you may want to try searching a range which starts at a smaller bond length.] Click on "Apply" and then on "Done." Repeat this procedure on the bond between the H atom and the other F atom. Save the fictitious molecule file. Answer CONTINUE when the program reminds you that you are violating valence rules with this molecule. (Why does this molecule violate valence rules?)
  2. Select MOPAC from the APPLICATIONS menu. Choose the "Rigid Search" calculation type and the "Grid" search type. Choose multiplicity equal to UHF. (You need to use UHF, the unrestricted Hartree Fock method, since the fictitious molecule has an odd number of electrons. The UHF method puts each electron in its own spin orbital so that they do not automatically have to be paired, as they are in RHF.) Type in "Geo-ok" under "Extra Keywords" under "Details." Run MOPAC.
  3. Choose VISUALIZER from the APPLICATIONS menu. Open the filename you previously created which now has a .Map extension. You will see two windows. The left window will contain the three-dimensional potential energy surface for the reaction, similar to that shown in Figure 2. It plots Potential Energy on the z-axis, as a function of the F-H and H-F distances in the fictitious molecule which is modelling the reaction. The right window displays the molecule, with its two bond distances labelled on the molecule, and its energy displayed at the top of the window.


    Figure 2. FHF5.Map showing two views of the potential energy surface of the head-on reaction between F(g) and H-F(g). The left view has the potential energy axis coming out of the screen towards the viewer, who looks into the x-y plane. The right view shows all three axes. Note the symmetry of the surface from the lower left-hand corner to the upper right-hand corner in the left view.
  4. Rotate the potential energy surface so that you can "see" its three-dimensional nature. An orientation which may be helpful is the one where the potential energy axis is pointing out of the screen at you (i.e. you see a two-dimensional plot of colored contours). Find the following regions and points:
    1. The reactant molecule in the reaction. Record its energy and bond length. Compare the bond length to that of H-F found in Exercise 1.
    2. The product molecule in the reaction. Record its energy and bond length. Compare the bond length to that of H-F found in Exercise 1.
    3. The transition state of the reaction (the activated complex). Record its energy and bond lengths.
    4. The region which represents the three separated atoms. Record its energy and location (range of bond lengths.)
    5. The point which represents the closest that the three atoms come together (out of the points calculated.) Record its coordinates.
    6. The reaction coordinate, i.e. the path that connects the reactants to the products and requires the smallest potential energy along the way. Sketch a picture of the contour plot and the reaction coordinate.

      Questions:

    7. How do the bond lengths in the transition state compare to those in the reactant (product)?
    8. Calculate the activation energy for the reaction.
    Approximate answers, using Figure 2:
    1. -51.8 kcal/mol, 0.90 A.
    2. -51.8 kcal/mol, 0.90 A.
    3. -18.6 kcal/mol, 0.90-1.05 A, 0.90-1.05 A.
    4. 90 kcal/mol, bond distances > 2.0 A.
    5. 0.75 A, 0.75 A.
    6. See, for example, P.W. Atkins, Physical Chemistry, 5th Edition, W.H. Freeman and Co., NY, 1994, page 953, figure 27.17, path c.
    7. Either the same, or else the transition state has slightly longer bonds - can repeat the calculation with higher resolution to decide.
    8. 33.2 kcal/mole

Exercise 3.

In this exercise you will compare the potential surfaces for the following four reactions:

F(g) + H-F(g) = F-H(g) + F(g) (1)
Cl(g) + H-F(g) = Cl-H(g) + F(g) (2)
Br(g) + H-F(g) = Br-H(g) + F(g) (3)
I(g) + H-F(g) = I-H(g) + F(g) (4)
  1. Follow the instructions from Exercise 2 to create the surfaces for reactions (2), (3), and (4). Figure 3 shows two views of a portion of the F-H-Cl surface.
    Figure 3. FHCl.Map, showing the potential energy surface for the head-on reaction between Cl(g) and H-F(g), shown in the two orientations of Figure 2. The axis marked "atom_dist0" varies the distance between the H and F atoms; that labelled "atom_dist1" varies the distance between the H and Cl atoms. Compare the symmetry of the left-hand view of the surface in Figure 2 to the asymmetry of the left-hand view of the surface in this figure.
  2. For each reaction:
    1. Record the energy and bond length(s) for the reactant molecule and the product molecule. Compare the bond lengths to those of the isolated molecules found in Exercise 1.
    2. Record the energy and bond lengths of the transition state. Compare the bond lengths to each other and to those of the isolated molecules.
    3. Calculate the forward and backwards activation energies of the reaction.
    4. Calculate the DH of the reaction. Is the reaction endothermic or exothermic?
  3. Construct a plot of X1-H bond distance vs H-X2 bond distance for the two bond distances in each of the four transition states.
  4. Discuss/explain the trends in all of the results.