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# Bonding in the Hydrogen Molecule

How is it possible for two atoms to be held together by a chemical bond? What is a chemical bond?

This exercise focuses on a very simple molecule: H2. Imagine two hydrogen atoms that are widely separated; the two atoms are initially so far apart that they do not interact with each other. Each hydrogen atom contains a single electron in the hydrogen 1s orbital.

What happens as the two atoms are brought close together?

The exercise below allows you to explore this process using an energy diagram, an electron density plot, and a virtual reality representation of the isosurface. One hydrogen atom is given the label "A" and the other "B". The controls allow you to specify the distance between the two nuclei (the isosurfaces are only available for a limited number of separation distances). An option is provided that plots small dots at the locations of the two nuclei, thereby allowing you to observe the positions of the two nuclei.

Start with the atoms at a separation of 4 angstroms (r = 4 angstroms) and gradually bring the atoms close together. At each distance, examine the electron density and isosurface plots. Identify the energy of the system on the energy plot. When the left mouse button is pressed while the cursor is over the energy plot, the position of the cursor on the plot is displayed.

There are three sets of interactions in this system:

1. Repulsion between the two electrons (one interaction of this type)
2. Repulsion between the two nuclei (one interaction of this type)
3. Attractions between each electron and nucleus (four separate interactions of this type)

 View:   Hydrogen A       Hydrogen B           Distance between nuclei >" />       0.2000.3000.4000.4500.5000.5500.6000.6500.7000.7380.8000.9001.0001.1001.2001.3001.4001.5001.6001.8002.0002.2002.5003.0004.0005.000 angstrom

Answers the following questions in terms of these interactions using your observations of the energy, electron density, and isosurface plots.

1. When r = 4 angstroms, what is the energy of the system? How does this energy relate to the energy of the hydrogen 1s orbital?
2. How close must the two nuclei be for there to be a significant interaction between the two atoms?
3. When the atoms are far apart, one electron is localized primarily around one nucleus and the other electron is localized primarily around the other nucleus. At what separation distance does this localization disappear? When the nuclei are closer than this distance, each electron spends equal amounts of time around each nucleus.
4. Set r to 0.900 angstroms and carefully examine the orbital. Describe the shape of the orbital. Where is the electron density concentrated? Why do the electrons behave this way? Is this state more stable or less stable than two isolated hydrogen atoms?
5. The H2 molecule will adopt the lowest energy state. In this lowest energy state, What is the separation distance of the two nuclei? (Report the answer to two decimal places.) This distance is called the bond length.
6. The bond dissociation energy for the H-H bond is the energy required to break the H2 molecule (in its most stable geometry) into separate hydrogen atoms. What is the bond dissociation energy for the H-H bond? Explain how this energy can be obtained from the energy diagram.
7. (The experimental bond dissociation energy for H2 is 436 kJ/mole. The data plotted below was obtained from relatively limited ab initio computations. Consequently the bond dissociation energy obtained from the energy plot will differ somewhat from the experimental value.)
8. Why does the energy of the system rise very sharply when r becomes very small? Where is most of the electron density? Or perhaps one should ask: Where isn't most of the electron density?

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