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Bonding to Hydrogen

The simplest molecule, made for connection

Roald Hoffmann

A Diatomic Molecule

Chemistry and I progressed; it took chemistry a good 75 years from Lavoisier’s time to have the macroscopic compounds—there at the beginning, with us today—be joined by a realization of an underlying microscopic reality, imagined well before it was proven, that of molecules. And it took another 65 years (now we’re circa 1925) for the new quantum mechanics to be created, explaining the why and wherefore of the molecules of dihydrogen (a nomenclature I will use when I need to distinguish hydrogen molecules from hydrogen atoms). In my education, I made that transition from compounds to molecules, much as chemistry did. Except I did it in three years instead of 140. I encountered the molecule, more precisely the quantum mechanical treatment of H2, in a class George Fraenkel taught, and beautifully so, in my last year at Columbia College.

Fraenkel took us through the first calculation on H2 by Heitler and London, in 1927, a calculation parlayed by Linus Pauling into a general theory of covalent bonding. By this time the dissociation energy of H2 (the strength of the bond, the energy needed to take it apart into two hydrogen atoms) was known. It was 4.48 electron volts (eV) per molecule, 104 kilocalories per mole (kcal/mol). If that doesn’t touch you, let’s begin with the fact that a mole of H2 (roughly 22 liters of it in gaseous form at room temperature) has a mass of 2.0 grams. Not much, that’s why it was used in airships. Kcal/mol? To heat a liter (about a quart, 1.057 quarts to be exact) of water from room temperature to boiling (a real-life operation most of us, even men, have done) takes about 80 kcal. That should help—to knock 2 grams of hydrogen molecules into hydrogen atoms takes about the same energy as to heat one and a quarter liters of water to boiling. Except, don’t try it on your stove—remember the Hindenburg airship.

2012-09MargHoffmannFB.jpgClick to Enlarge ImageThe energy of the hydrogen molecule as a function of distance is described by a “potential energy curve” shown in the figure below, a graphical depiction of how the chemical potential energy of the molecule varies with separation of the hydrogen atoms (actually their nuclei) in the molecule from each other. The depth of the well relative to the separated atoms is the dissociation energy I described above. But any molecule is a quantum mechanical entity; so the molecule, in a way as a consequence of Heisenberg’s uncertainty principle, does not sit still at the minimum of the potential energy curve. The molecule vibrates, the vibrations of the molecule are quantized—and in its lowest energy state the hydrogen nuclei retain some motion (in a way like a pendulum but less deterministically so) around the “equilibrium distance.” Sometimes they are a little closer, sometimes a little farther apart, on the average they are ~0.74 × 10–8 centimeter, 0.74 Ångström (Å) from each other. We call that the bond distance.

The bond distance in the H2 molecule and its dissociation energy were known by the time the new quantum mechanics came. Heitler and London got a dissociation energy of 3.14 eV, an equilibrium distance of 0.87 Å. Not too great (compared with experiment) but a remarkable result: For the first time quantum mechanics “explained” the existence of a molecule. Which classical mechanics coupled with electrostatics, try as it might, couldn’t.

One could not solve the Schrödinger equation, the wave equation that describes all matter, exactly for H2, but the path down a road of increasingly accurate approximations to the exact solution seemed beautifully logical and enticing to this young apprentice. Fraenkel took us through it first of all by another method, called the molecular orbital (MO) method, pioneered by Friedrich Hund and Robert S. Mulliken. A molecular orbital is a combination of atomic orbitals, an approximate way to describe the location of electrons in a molecule—I will show you one soon. This method eventually dominated chemical thinking from the 1950s through today, but initially gave a poorer description of the H-H bond in H2. Yet both the MO and the Heitler-London methods (expanded into Pauling’s “valence bond” [VB] approach) could be systematically, logically improved. We followed and understood that path in our class, culminating in a remarkable 1933 calculation by H. M. James and A. S. Coolidge, using hand-cranked mechanical calculators, that matched experiment.

I would like to show you the molecular orbitals of H2, because (a) they’re important, and (b) I can’t escape them; they bring to me new chemistry at roughly 25-year intervals. The two 1s orbitals of the individual atoms combine in in-phase and out-of-phase fashion to give molecular orbitals called σg and σu*, shown in the figure at the top of the next page.

2012-09MargHoffmannFC.jpgClick to Enlarge ImageThe σ and the subscripts and superscripts on it are labels, symmetry labels; what matters is that σg has no node between the nuclei, while σu* does. That puts σg low in energy, σu* high. And, importantly, σg is a “bonding” orbital, if occupied (as it is in H2), the electrons in it bring the atoms together, whereas σu* is an antibonding orbital, any electrons in it (there are none in an unperturbed H2 molecule, at least in the simpolest analysis) pushing the nuclei apart. Interesting that the big guys, the massive nuclei, move where the small electrons tell them to move.

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