Series wave functions for the helium atom

Abstract

Series methods for solving differential equations have been applied to the Schrodinger equation for helium. Formally exact solutions have been obtained for singlet S states. The solutions are expressed as multipole expansions in the angle between the electron position vectors r1 and r2 at the nucleus. The radial functions cannot be expressed as power series in r1 and r2, but as series involving powers of r1, r2, log r1, and log r2. Recurrence relations, together with conditions for smoothness at r1 = r2 and proper behavior at the nucleus, determine most of the expansion coefficients. Simple, exact expressions for infinitely many of the coefficients, namely those for terms involving ri1rj2 with i + j = 0, 1, and 2, have been determined. Coefficients not determined in the recursion process (of which there are infinitely many for each multipole component) can have arbitrary values in the formal solution. They are determined by the additional requirement that the wave function should have a finite square integral. An observation that the eigenfunction should be asymptotically separable in r1 and r2 for large values of these variables leads to approximate relations among the arbitrary coefficients, reducing the number of undetermined parameters to one per multipole component. Estimates for these remaining coefficients can be made easily. All approximations can be improved systematically to obtain arbitrarily high accuracy. The possibility of obtaining the exact solutions is considered.