Brief history of the helium problem

Hylleraas. These results showed that the new quantum theory is a reliable basis for the computation of the spectra of many-electron systems. 

A new field of research opened with the experimental work of Madden and Codling (1963), who proved the existence of autoionizing states embedded in the continuum. High accuracy calculations of resonances were subsequently reported by many researchers. We mention, e.g., the work by Ho (1981), Bhatia and Temkin (1984), Lindroth (1994), Muller et al. (1994), and Burgers (1995). 

A reactionary movement started with the work of Leopold and Per-cival (1980). Using modern semiclassical techniques these authors were able to show that the old quantum mechanics was not so bad after all. Improving the old theory with the help of Maslov indices and variational techniques, Leopold and Percival showed that the old quantum theory yields results for the ground state and excited states of helium that are within the experimental accuracy achieved by the 1920s. Thus, Leopold and Percival turned the failure of the old quantum theory into a success, since the accuracy of the semiclassical theory improves with increasing quantum numbers and turns out to be a very useful tool for the computation of highly excited states. 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

For fixed N, the second electron describes a Rydberg series of states converging to an ionization threshold at 

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