Ground-state of He-like ions

let us consider the helium ground state. As a first step, we solve the Hartree-Fock equation 

To go further, we must evaluate the higher-order corrections. Setting a = 6 = Is in 

Partial wave contributions (a.u.) to the second-order ground-state energy in helium. 

we obtain the following simple expression for the second-order energy  

Taking the radial wave functions and energies for states n from the B-spline basis set, we may easily carry out the double sums in (128). The partial-wave contributions to from terms in square bracket are listed in Table 2. These terms fall-off approximately as L for large L and may easily be extrapolated. We find E = —0.0373736 a.u., leading to a binding energy of -0.8990800 a.u., differing from experiment by 0.5%. 

---

Lowdin, P.-O., and Redei, L., Combined use of the methods of superposition of configurations and correlation factor on the ground states of He-like ions."... 

Interesting results were found for the partial wave expansions. If one expands the wave function of the ground state of He-like ions as... 

---