Signature of chaos in high-lying doubly excited states of the helium atom

We examined nearest-neighbor spacing (NNS) statistics in doubly excited states of helium near the double ionization threshold. Using the Brody parameter q to measure the NNS distribution between the regular Poisson distribution (q=0) and the chaotic Wigner distribution (q=1), we showed that for levels near the N=20 threshold of He⁺, or at about 0.13 eV below the double ionization threshold, the NNS distribution has q=0.66. The result shows the slow approach of the NNS of helium energy levels towards the Wigner distribution vs the excitation energy. Using an s-wave model where the angular momentum of each electron is restricted to zero, we also examined the NNS for levels up to the N=30 threshold of He⁺. We showed the gradual increase in q as the excitation energy is increased. To generate the theoretical data needed for the NNS analysis, we have used the hyperspherical close-coupling method, with the recently proposed diabatization of potential curves and the truncation of channels, to greatly reduce the complexity in the calculation. We also investigated the dependence of q vs the nuclear charge in the same scaled energy region, and for different symmetries, to assess their relation with the rate of approaching the q=1 Wigner distribution.