The one- and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufficiently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for $Z<1$ a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge $Z_c\approx 0.911028...$, at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system.

The critical screening parameters for one-electron systems screened by Hulthén, Debye–Hückel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm -scaling law. A thorough comparison of the critical screening parameters for the ground and the first p -wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system-bound and pseudo-continuum eigenenergies for s - and non- s -wave states are shown to follow the quadratic and linear laws, respectively. The variation of the corresponding wave functions is analyzed in detail. For systems with non-zero orbital angular momenta, the bound states convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l -dependent power law. It is further shown that the different asymptotic behaviors of the resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.

We develop an efficient numerical method to directly calculate the critical screening parameters for one-electron systems with Hulthén and Debye–Hückel screened Coulomb potentials (SCPs). Compared to indirect methods, which locate the critical screening parameters via searching the potential parameters with near-zero energy, the method developed in this work directly calculates the critical screening parameters as eigenvalues of a generalized eigenvalue problem. This feature allows us to simultaneously determine the critical parameters for bound states from low-lying excitation to high-lying Rydberg limit with high accuracy. The method is applied to SCPs to investigate the asymptotic behavior of critical screening parameters as the principal quantum number n approaches infinity. It has been shown that the critical screening parameters in Hulthén and Debye–Hückel potentials follow the 2n ⁻² and 4n ⁻²/π asymptotic laws, respectively, and that the orbital angular momentum affects the higher-order coefficients linearly.

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.