Asymptotic behavior of two-electron expectation values in two-electron excited states

Singly-excited states of the two-electron atom cease being bound when Z → 1 (from above), the outer orbital becoming infinitely diffuse. The asymptotic relations lim Z → 1 ⁡ ( Z − 1 ) k 〈 ( 1 s n s ) 1 , 3 S | r 12 k | ( 1 s n s ) 1 , 3 S 〉 = 〈 ( n − 1 ) s ( 0 ) | r k | ( n − 1 ) s ( 0 ) 〉 , where k = − 1 , 1 , 2 , 3 , ⋯ , are demonstrated to hold. Here, ( n − 1 ) s ( 0 ) is a hydrogenic s orbital with principal quantum number ( n − 1 ) . New, more nuanced light is shed on the already challenged dogma that the Pauli principle keeps the electrons further apart in the triplet than in the corresponding singlet.