Extracting Potentials from Spectra

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Science  19 Jun 2009:
Vol. 324, Issue 5934, pp. 1526-1527
DOI: 10.1126/science.1175751

For most elements, we know whether they can form a diatomic molecule, especially for light atoms that have few electrons and can be treated readily by theory. But for one such light element, surprises are still in store. For most of the 20th century, experimental and theoretical studies agreed that the beryllium dimer (Be2) did not exist. The Be atom has filled electron shells and—like the inert gases such as helium—was expected to form at most a weak van der Waals dimer at very large internuclear distances. Yet, as shown experimentally by Merritt et al. on page 1548 of this issue (1), Be2 does exist and has a relatively short bond (2.45 Å), relative to the anticipated van der Waals complex with a bond length of about 5 Å. Its unusually flat potential curve limits the number of vibrational levels and provides the rare opportunity to study the highest vibrational state of a molecule just at its dissociation limit.

Experiments with beryllium are difficult because the metal is refractory (it has a low vapor pressure even at very high temperatures) and because beryllium-containing compounds are generally extremely toxic. However, Be vapor can be created through laser ablation of a Be metal target. Rapid cooling of the vapor during supersonic expansion through an orifice into vacuum allows preparation of the dimer. The rovibrational states of the dimer are then probed with a double-resonance method: One laser excites the molecule into an excited electronic state where the atoms are still bound; stimulated emission pumping (2) by a second laser returns the molecule back into each of the bound vibrational levels of the ground state.

Shallow potentials with deeper implications.

The potential energy function for Be2 as a function of interatomic distance r was determined by Merritt et al. from a fit to the experimental observations. The levels become more congested as the energy nears the dissociation limit De. The bound vibrational energy levels and the square of the vibrational wave functions were calculated by LeRoy with his program LEVEL (6).

The data analysis performed by Merritt et al. is noteworthy because it allows a better connection to theory than standard methods. Vibration-rotation energy levels are usually reduced to spectroscopic constants that are parameters in a power series expansion that uses the relevant quantum numbers of the states (3, 4). However, an excessively large number of expansion terms are needed, particularly for a potential with an unusual shape such as Be2, and these fitting parameters have lost their physical meaning. In contrast, a parameterized potential function (see the figure) requires far fewer fitting parameters and makes a direct connection with ab initio quantum chemistry.

Merritt et al. adopted a more powerful analysis method that bypassed traditional constants in favor of a parameterized potential energy function obtained from a direct fit of the energy levels using the vibration-rotation Schrödinger equation. LeRoy has been one of the pioneers of this modern quantum mechanical approach, and the authors used his freely available computer code (5, 6).

With only eight electrons, Be2 has attracted and challenged quantum chemists for nearly 80 years. Ab initio quantum chemistry solves the electronic Schrödinger equation to obtain the interaction energy for a series of molecular geometries. Within the Born-Oppenheimer approximation, which separates fast electronic motion from slower vibration-rotation nuclear motion, these electronic energies are used to construct the potential energy function that is then used to solve the vibration-rotation Schrödinger equation. It is now even possible to deal with the breakdown of the Born-Oppenheimer approximation in both experiment (7) and ab initio theory (8).

Quantum chemistry attributes the formation of the Be–Be bond and the unusual potential shape to the inclusion of more and more p character in the valence orbitals as the atoms approach each other (9). The p orbitals are more directional than the s orbitals that describe the isolated atoms. The comparison between the experimentally derived potential of Merritt et al. and modern high-quality ab initio potentials such as those calculated in (9, 10) is reasonable at the moment but definitely not perfect (11), with noticeable differences in the internuclear distance and in the dissociation energy.

It is experimentally challenging to measure the entire range of vibrational levels and locate the last bound level at v = 10, which has a binding energy of only a few wave numbers. Indeed it is even possible that there is another vibrational level with v = 11 that is bound by a tiny fraction of a wave number. As can be seen from the square of the wave function, which gives the probability distribution, the molecule spends nearly all of its time at a large internuclear separation of 7 Å for v = 10. The application of stimulated emission pumping and the availability of a suitable excited electronic state allowed Merritt et al. to map out the Be2 levels.

Spectroscopic observations often focus on the lowest few vibrational levels of a potential because they are easier to observe and calculate, but modern computational and experimental methods—such as interactions seen in cold atom trapping (12)—can provide information on the last few levels near dissociation. In some cases [for example, MgH (13)], it is only the combination of modern potential fitting methods with high-quality observations that enables the last few bound levels to be located in the forest of stronger lines. The full characterization of all bound levels with experimental precision for small molecules such as water is still a distant goal, but the first steps with, for example, H2 (14), MgH (13), and now Be2, are important landmarks on the way.

References and Notes

  1. See
  2. I thank R. J. LeRoy for making the figure and for his comments. Financial support was provided by the UK Engineering and Physical Sciences Research Council.
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