Chemical Bonding in the H₂ Molecule

Energy of Two Hydrogen Atoms

In the H₂ molecule, chemists talk about the two hydrogen atoms being held together by a chemical bond.
What is a chemical bond?

The key concept in understanding chemical bonding is energy. The collection of atoms will tend to exist in the geometry that produces the lowest energy.

For two hydrogen atoms, there are three sets of interactions that determine the potential energy of the system.

1. The repulsion between the two nuclei, each with +1 charge. (one interaction of this type)
2. The repulsion between the two electrons, each with -1 charge. (one interaction of this type)
3. The attraction between an electron and a nucleus. (four interactions of this type)

The graph below shows the variation in the potential energy for two hydrogen atoms as a function of the distance ( r ) between the two nuclei.

Use the above energy graph to answer the following questions.

1. The energy of an isolated hydrogen atom is -13.60 eV. What is the energy of the system when the two hydrogen atoms are far apart?
2. How close must the two nuclei be for there to be a significant interaction between the two atoms?
3. Why does the energy of the system rise very sharply when r becomes very small?
4. The nuclei will adopt the separation distance that produces the lowest energy state. Because we think of these two hydrogen atoms as being bonded together, we call this distance the bond length. What is the bond length for the H₂ molecule?
5. The atoms are most stable when r equals the bond length (Question 4). Suppose one wished to separate the H₂ molecule into two widely separated H atoms. How much energy is required to bread the H-H bond? (This value is called the bond dissociation energy.)

The graph is based upon low-level ab initio calculations. For comparision, the experimental H-H bond length is 0.74 Å and the experimental bond dissociation energy is 436 kJ/mole or 4.52 eV.

H-H Chemical Bond

As explained above, a chemical bond is not a physical thing. It is simply a term chemists use to indicate that atoms sometimes stay close together (rather than moving far apart) because their energy is lower when they stay together.

For an isolated hydrogen atom in its ground state, the one electron in the molecule is described by the 1s wave function. We say the atom is "in" the 1s orbital. That terminology is unfortunate. It would be more accurate to say that the 1s wave function describes the state of the electron, that is, it describes the way the electron exists.

When two hydrogen atoms are bonded together ( r = 0.74 Å ), each electron is described by the wave function (that is, an orbital) for the chemical bond. In this case, the bond is a σ bond, so there is a σ wave function. This wave function allows each electron to spend time near each nucleus. In fact, an electron spends most of its time between the two nuclei, though the electron does occasionally visit the far side of each nucleus.

A simple mathematical approximation for the σ wave function is to simply add the 1s wave functions for each hydrogen atom.

The reason for this approximation is illustrated in the electron density map and isosurface viewer below.
The small black dots in the electron density map and the small white spheres in the isosurface viewer represent the positions of the nuclei of the two hydrogen atoms.

Use the check boxes to look at the 1s orbital for each of the individual hydrogen atoms. Then select both H 1s boxes to superimpose the two 1s orbitals. Notice that there is significant overlap of the two orbitals.

Next select the H₂ σ orbital. Notice the close similarity between the σ orbital and the sum of the two H 1s orbitals. The agreement is not exact, which is why treating σ as the sum of two H 1s orbitals is an approximation, but the two depictions are very similar.