Hund's rules and the interpretation and common misinterpretation of energy differences

With a Z expansion formalism which is exact in non-relativistic quantum mechanics it has been shown that for multiplets of neutral atoms and of many positive ions the state of the highest energy has the lowest expectation value for the interelectronic repulsion energy. This reversed order of the repulsion energy occurs for cases which obey Hund's first rule as well as for cases which obey the second of Hund's rules. It can be shown that the energy differences are in all cases dominated by the difference in electron nuclear attraction energy and not by the difference in electron repulsion energy.

The lowest ¹Π _u and ³Π _u states of the H₂ molecule have similar features. At many internuclear distances, including the equilibrium ones, the ¹Π _u state has the highest energy but the lower kinetic energy and electron repulsion energy but again the higher electron nuclear attraction energy. These results contradict clearly the usual theories for energy differences between spectroscopic states in atoms and molecules.