Using the asymptotic iteration method (AIM), we investigate the variation in the 1s energy levels of hydrogen and helium-like static ions in fully degenerate electron gas. The semiclassical Thomas–Fermi (TF), Shukla–Eliasson (SE), and corrected Shukla–Eliasson (cSE) models are compared. It is noted that these models merge into the vacuum level for hydrogen and helium-like ions in the dilute classical electron gas regime. While in the TF model, the hydrogen ground state level lifts monotonically toward the continuum limit with an increase in the electron concentration; in the SE and cSE models, a universal bound stabilization valley through the energy minimization occurs at a particular electron concentration range for the hydrogen-like ion which for the cSE model closely matches the electron concentrations in typical metals. The latter stabilizing mechanism appears to be due to the interaction between plasmon excitations and the Fermi length scales in the metallic density regime. In the case of helium-like ions, however, no such stability mechanism is found. The application of the cSE model with electron exchange and correlation effects reveals that the cSE model qualitatively accounts for the number density and lattice parameters of elemental metals within the framework of free electron assumption. According to the cSE model of static charge, screening a simple metal–insulator transition criterion is defined. The effect of the relativistic degeneracy effect on the ground state energy of the hydrogen atom is studied. It is shown that the ground state energy level of the hydrogen atom also undergoes a collapse at the well-known Chandrasekhar mass limit for white dwarf stars.
