Bound and resonance states near the critical charge region in two-electron atoms

The critical nuclear charge $Z_c$ of the two-electron atoms below which the ground state transforms into a shape resonance and the critical stability of the system around $Z_c$ have been well established. However, the behavior of the shape resonance below $Z_c$ is still a mystery. By employing the complex-scaling method using Hylleraas configuration-interaction basis functions, we trace the trajectory of the shape resonance from $Z_c$ down to a very small nuclear charge. It is shown that at specific values of $Z$ far below $Z_c$ the resonance crosses over higher-lying one-electron thresholds, and when $Z$ is decreased below 0.316, the shape resonance lies above the three-body breakup threshold. We finally show that the imaginary part of the resonance energy at small nuclear charges can be modeled by the dispersion relation with a high-order Padé approximant correction.

Figure 1

The bound and resonance energies of the two-electron atoms as a function of nuclear charge $Z$. (a) Real part of the energies and (b) imaginary part of the energies. In each plot, the inset shows the variation of $Z$-scaled energies $E/Z^2$ as a function of $Z$. $Z\left(n\right)$ in (a) represents the threshold energy of a one-electron atom with principal quantum number $n$.

Figure 2

The complex rotated eigenenergies of the system Hamiltonian at different values of rotational angle $\theta$ and scaling parameter $\alpha$. Solid lines in (a), (c), and (d) are drawn at fixed values of $\theta$ with varying $\alpha$, while in (b) they are at fixed values of $\alpha$ with varying $\theta$. The resonance energy is expected to be located at the center of the rotated spectra. (a) $Z=0.80$ with $\theta =0.72–0.82$ and $\alpha =1.02–1.30$ both at intervals of 0.02, (b) $Z=0.80$ with $\alpha =1.02–1.24$ and $\theta =0.70-0.84$ both at intervals of 0.02, (c) $Z=0.381$ with $\theta =0.58–0.68$ and $\alpha =0.70–0.98$, and (d) $Z=0.316$ with $\theta =0.44–0.54$ and $\alpha =0.68–0.94$. The dashed line in (d) demonstrates the $E=0$ three-body breakup threshold above which a well-isolated resonance is clearly observed.

Figure 3

Comparison among the resonance position, the autoionization threshold, and the top of the model potential barrier. $E_r$ is the real part of the complex resonance energy, $E_{Z(n=1)}$ is the ground-state energy of the one-electron atom, and $V_{\text{top}}$ represents the absolute height of the potential barrier including the threshold energy. The inset shows the corresponding $Z$-scaled energies as a function of $Z$.

Figure 4

Trajectory of the eigenvalue of the two-electron atoms in the complex-energy plane with decreasing $Z$. States for $Z<0.3162$ are expected to be the positive-energy shape resonances with central positions lying above the three-body breakup threshold. The inset shows the trajectory of the $Z$-scaled eigenvalues.

Figure 5

Fitting of the numerical calculation of the imaginary part of the resonance energies as a function of $1-Z/Z_c$. The fitting parameters obtained by Karr [23] and Dubau and Ivanov [28] are given in Eqs. (11) and (14), respectively. Padé$[M,N]$ represents the improved asymptotic law of Eq. (15), which includes the correction of Padé$[M,N]$ approximants.