Where is the Chaos in Two-Electron Atoms?

The classical dynamics of two electrons moving in the presence of a single, fixed, nucleus is largely chaotic. What is the effect of this on the spectra of two-electron atoms? Regular sequences of bound states and resonances are seen experimentally to quite high quantum numbers. Scattering cross sections are smooth, with well understood threshold effects in both theory and experiment. Where are the chaotic levels and/or chaotic scattering regimes? Or, are atomic systems too quantum-like to show any manifestations of the underlying classical dynamics? To investigate these questions, and the corresponding problem of semiclassical quantization, we have investigated the classical and quantum dynamics of a model two-electron atom wherein each electron is confined to one-dimensional motion on opposite sides of the nucleus. Periodic orbits have been enumerated, and a simple coding found (as has been independently done by Kim and Ezra, Richter and Wintgen). This allows application of the Gutzwiller trace formula to relate classical and quantum spectra. Fully quantum computations have been carried out using the method of complex coordinates to allow variational computation of resonances. A regime is found, beginning at approximately the n = 10 threshold, where the resonance eigenvalues behave like a two-dimensional gas, and a Pechukas-type theory of their motion is developed. This is one analogue of classical chaos. Further, near n = 30, estimates indicate that the typical spacing of the real parts and the average magnitude of the imaginary parts of the “gas” of resonances are approximately equal. In this regime we predict the onset of quantum chaotic scattering.