Precise criterion for excited states of the lithiumlike systems

By using the accurate inequalities of the electron density at the nucleus and radial expectation values for atomic systems derived by Gálvez and Porras, we separately construct two sets of precise criterion of the wave function, they are another precise criterion independent of energy to test quality of the wave function. In this paper, we use the excited states of lithiumlike systems as research objects and examine the upper bound of the electron density at the nucleus and radial expectation value <r> for the lithiumlike systems 1s²3s state, R₆ defined by the upper bounds of the electron density at the nucleus and R₄ defined by the upper bounds of radial expectation value <r> for the lithium atom Is²3s state in different calculation methods are also tested, at the same time we check up the upper bound of the electron density at the nucleus and radial expectation value <r> for double excited states of the lithium atom. Research has shown that these two sets of inequalities can be used as precise criterion for wave function of energy states of atomic systems.