Mathematical Challenges from Theoretical/Computational Chemistry

The Born-Oppenheimer approximation dates from the 1920s, and the entire notion of molecular structure can be based upon it. It is thus a surprise that significant qualitative physics has been ignored by most chemical physicists in applying the Born-Opp enheimer approximation to systems with degenerate electronic states. The basic idea behind the Born-Oppenheimer approximation is that nuclei move much more slowly than electrons. Thus, the Schrödinger equation for electrons can be solved at fixed n uclear configuration and the resulting energy can be used as a potential for studying the motions of the nuclei themselves.

Generally, when nuclear motion itself is quantized, one assumes the usual Schrödinger equation with a classical scalar potential for the nuclear motions. This has proved valid for systems that do not have significant electronic degeneracy. A seri ous mathematical problem is the uniqueness of the wave function for the nuclei. The Born-Oppenheimer approximation really assumes a single path for the slowly moving nuclei. If there is an electronic degeneracy, topologically distinct paths may connect two different positions on the same electronic surface. Thus, in addition to the phases that one develops for the quantum dynamics through the simple scalar potential dynamics, there is an additional topological phase. The existence of this topological phase, which depends on the path between two points, has been known since at least the 1950s, when Longuet-Higgins studied it in the context of Jahn-Teller distortions. Only in recent years has its significance been truly appreciated, however. One of th e leaders in bringing out the significance of topology in quantum molecular dynamics was M. Berry. However, it was appreciated somewhat earlier by Truhlar and Mead that this topological phase plays a role in chemical reactions. Indeed it is important ev en in the most fundamental of chemical reaction problems, the H + H reaction. Very recently, the discrepancy between experimental results for H + H and large-scale computat ions of the scattering cross sections was shown to arise from neglect of this topological phase.

Box 4.3

For problems with small amounts of degeneracy, the topological phase is easy to handle with little mathematical sophistication. Either a trajectory encircles a conical intersection (of Born-Oppenheimer energy surfaces) or it does not, leading to two va lues of the phase. This encircling of singularities can be described by using the idea of a gauge potential. With higher degeneracies, however, very difficult topological problems may be encountered since many surfaces can make avoided crossings in many locations. The paradigm of such complicated topology problems may well be metal clusters. For metals in the thermodynamic limit, there are numerous energy levels corresponding to the excitation of electrons just below the Fermi sea to just above it. S ince the electronic levels are highly delocalized, these energy changes are quite small and the energy surfaces are close together. The actual dynamics of the nuclei must involve the coupling of several surfaces. There are many possible interchanges of the metallic ionic cores, and complicated topologies can result.

Another place in which topology enters is when an underlying approximate wave function is built up out of many degenerate electronic wave functions and the dynamics of electronic excitations is studied. The paradigm for this is the recent interest in r esonating valence bond descriptions of metallic and superconducting materials. Here, reorganization of the different valence bond structures as an excited electron or hole moves around gives rise to topological phases and gauge fields. It has been argue d that these effects are at the heart of the new high-temperature superconductors and represent a real breakdown of the traditional band structure picture of metals. Most models studied by physicists, however, have been very simple, and it will be necess ary to understand how the topological phases arise in completely realistic electronic structure calculations if one is to make predictions of new high-temperature superconductors on the basis of these ideas.

NAS Home Page | NAP Home Page | Reading Room | Report Home Page