Geometric properties of the first singlet S-wave excited state of two-electron atoms near the critical nuclear charge

The geometric structure parameters and radial density distribution of 1s2s ¹S excited state of the two-electron atomic system near the critical nuclear charge Z_c were calculated in detail under tripled Hylleraas basis set. Contrary to the localized behavior observed in the ground and the doubly excited 2p²³P^e states, for this state our results identify that while the behavior of the inner electron increasingly resembles that of a hydrogen-like atomic system, the outer electron in the excited state exhibits diffused hydrogen-like character and becomes perpendicular to the inner electron as nuclear charge Z approaches Z_c. This study provides insights into the electronic structure and stability of the two-electron system in the vicinity of the critical nuclear charge.