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 ${\mathrm{He}}^{+}$, or at about $0.13\mathrm{eV}$ 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 ${\mathrm{He}}^{+}$. 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.

Figure 1

(Color online.) $\mathrm{He}\left({}^{1}P^0\right)$ potential curves converging to $I_4$ and the lowest curve converging to $I_5$. The thick colored curves are adiabatic, the thin black curves are diabatic.

Figure 2

(Color online.) ${}^{1}S^e$ diabatized potential curves for 3D helium between $I_{12}$ and $I_{24}$ ionization thresholds. The thick red curves correspond to the lowest curves from each manifold. They have the approximate quantum numbers $K=N-1$.

Figure 3

(Color online.) Lowest potential curves from each manifold up to $N=24$ for ${}^{1}S^e$ 3D helium (that is, the red curves in Fig. 2), plotted as effective quantum numbers $N_{\mathrm{eff}}\left(R\right)=\sqrt{-2∕U\left(R\right)}$ vs $R^{1∕2}$.

Figure 4

(Color online.) Close-up of the ${}^{1}S^e$ hyperspherical potential curves in the energy region near the $I_3$, $I_9$, and $I_{18}$ thresholds.

Figure 5

(Color online.) ${}^{1}S^e$ energy levels below $I_{18}$ threshold from one-channel calculations for $N=18$ to $N=22$, in columns (a)–(e), respectively. Column (f) groups all “noninteracting" levels from (a)–(e) together and column (g) represents the results from coupled-channel calculations.

Figure 6

${}^{1}S^e$ potential curves for $s$-wave model of helium $(Z=2)$ from $N=21$ up to $N=40$.

Figure 7

(Color online.) NNS distribution from HSCC calculations in the energy range $I_{25}<E<I_{30}$ for an $s$-wave ${}^{1}S^e$ model two-electron atoms with nuclear charges $Z=\infty$ (i.e., independent electrons) (a), $Z=6$ (b), and $Z=2$ (c). The Poisson and Wigner distributions are shown as the dotted red and dashed blue curves, respectively.

Figure 8

(Color online.) NNS distribution for ${}^{1}S^e$ states from real 3D HSCC calculations for the energy range $I_{14}<E<I_{19}$. The Poisson and Wigner distributions are shown as the dotted red and dashed blue curves, respectively.

Figure 9

(Color online.) Comparison between the ${}^{1}S^e$ potential curves from $N=11$ to $N=20$ for the $s$-wave model of helium and $K=N-1$ potential curves of real 3D helium.