Numerical Solution of the Two-Electron Schrödinger Equation

Numerical solutions to the $S$-limit equations for the helium ground state, excited triplet state, and the hydride-ion ground state are obtained with the second and fourth difference approximations. The results for the ground states are superior to previously reported values. The coupled equations resulting from the partial-wave expansion of the exact helium atom wave function are solved giving accurate $S,P,D,F$, and $G$ limits. The $G$ limit is -2.903 51 a.u., compared to the exact value of the energy of -2.903 72 a.u.