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.

The hyperpolarizabilities of the hydrogenlike atoms in Debye and dense quantum plasmas are calculated using the sum-over-states formalism based on the generalized pseudospectral method. The Debye-Hückel and exponential-cosine screened Coulomb potentials are employed to model the screening effects in, respectively, Debye and dense quantum plasmas. Our numerical calculation demonstrates that the present method shows exponential convergence in calculating the hyperpolarizabilities of one-electron systems and the obtained results significantly improve previous predictions in the strong screening environment. The asymptotic behavior of hyperpolarizability near the system bound-continuum limit is investigated and the results for some low-lying excited states are reported. By comparing the fourth-order corrected energies in terms of hyperpolarizability with the resonance energies using the complex-scaling method, we empirically conclude that the applicability of hyperpolarizability in perturbatively estimating the system energy in Debye plasmas lies in the range of $[0,F_{\text{max}}/2]$, where $F_{\text{max}}$ refers to the maximum electric field strength at which the fourth-order energy correction is equal to the second-order term.

The critical screening parameters (λ _c ) of one-electron systems in circular states under short-range screened Coulomb potentials (SCPs) are investigated using the generalized pseudospectral method. High-precision values of λ _c are obtained for bound states with the principal quantum number n up to a thousand. The polynomial fittings based on the numerical values indicate that the high-Rydberg limits of the n ²-scaled critical screening parameters, i.e. n 2 λ c , are equal to 2 / e , 1.295 220 475 783 830 , and 0.543 564 037 493 402 , respectively, for the Debye–Hückel, Hulthén, and exponential cosine SCPs. It is surprisingly found that these high-Rydberg limits can be well-reproduced from Bohr’s correspondence principle together with the semi-classical model of the hydrogen atom. We further show that both the present numerical calculations and the semi-classical analysis can be extended to other short-range potentials such as the generalized exponential, Lennard–Jones, modified Pöschl–Teller, shifted Morse, Woods–Saxon, and Manning–Rosen potentials that attract wide interest in atomic, molecular, and nuclear physics.

The kinetically and atomically balanced conditions have been extensively used in various basis-set expansion methods to remove the nonphysical spurious states appearing in the numerical calculation of the Dirac equation; however, they are generally not applicable for methods in discrete variable representation. In this paper we show that these conditions can be conveniently introduced into the generalized pseudospectral (GPS) method. Four types of balanced condition, the mono kinetic balance (MKB), dual kinetic balance, mono atomic balance (MAB), and dual atomic balance, are incorporated into the GPS method to eliminate the spurious states. Numerical calculations for a variety of bound states of H-like ions in point-charge models are compared with the analytical solutions to demonstrate the accuracy and efficiency of the developed methods. The application to highly charged ions with extended nuclear models is performed to show the flexibility of the balanced GPS methods in practical atomic structure calculations. It is concluded that the MAB-GPS and MKB-GPS methods, which are both free of any spurious states, show better performance and simpler implementation than the others in solving the Dirac equation with the potential in point-charge and extended nuclear models, respectively. The balanced GPS methods developed in this paper provide a useful tool for accurately solving the one-electron Dirac equation and efficiently constructing the multielectron relativistic wave functions.