2-ELECTRON ATOMS - KINOSHITA EXPANSION

Kinoshita has pointed out that the famous Hylleraas expansion, =Al,m,n slumtn, is unable to furnish a formal solution to the Schrödinger equation for the two-electron atom. He further noted that the source of the difficulty is the arbitrary restriction of the indices l and m to positive values. The Kinoshita variables s=s, p=us, and q=tu alleviate this difficulty, and a Kinoshita expansion =Cl,m,n slpmqn seems capable of furnishing formal solutions. It is shown here for S states that Cl,m,n=0 for all Cl,m,n with n>m. This result is equivalent to the statement that negative powers of u are not needed. This result, when applied to the 1S1 ground state, allows the determination of all coefficients of the form C2,21/4+1,n. The implied infinite series are summed and explicit closed-form expressions are found which involve a novel logarithmic term. Conditions which must be imposed on that part of constructed from C2,21/4,n are discussed. © 1979 The American Physical Society.