Use of the Fock expansion forstate1wave functions of two-electron atoms and ions

The exact representation of a two-electron wave function near the origin is the Fock expansion, i.e., a double summation over powers of R and of lnR [where R≡($r_1^2$+$r_2^2$ ${)}^{1/2}$] with coefficients dependent on the five remaining angular variables. Using a representation of hyperspherical harmonics, we present here the first numerical solution of the equations for the Fock coefficients. We present also a general procedure for matching a linear combination of Fock-series solutions onto a basis of adiabatic hyperspherical functions at a matching radius $R_0$. This matching procedure ensures that the proper asymptotic boundary conditions are satisfied. Exploratory numerical results are presented for ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ wave functions of He and ${\mathrm{H}}^{\mathrm{-}}$ in which four Fock-series solutions are matched onto the lowest ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ adiabatic hyperspherical wave function at a matching radius near the first antinode in the adiabatic wave function.