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 Pad\'e approximant correction.