In this work we analyze the spectrum of the iso-electronic series of the helium atom by using the hypespherical adiabatic method. This method allows the study of the energy levels of atomic systems by means of a set of potential curves, similarly to the Born-Oppenheimer approach for molecular systems. The potential curves are defined with respect to a unique radial variable, regardless the number of the system electrons. As a result the analysis and classification of the energy levels proceeds in a simple and intuitive way, which is not observed within the variational and Hartree-Fock methods. The objective of this research is to describe the behavior of the helium-like systems ground state with respect to the nuclear charge value. Besides the hyperspherical method simplification of the results analysis, it is an ab-initio process, whose accuracy is only limited by the truncation of the coupled equation number. Within its simplest approximation, in which all couplings are disregarded, the error obtained for the ground state energy is inferior to 1% and with the introduction of the diagonal coupling the error drops to only 0.3%. Further improvement is obtained with the non-adiabatic couplings, reaching precisions of the order of 10-3 %.
Atomic spectra; potential curves; hyperspherical method; two electrons
