We compute the electrostatic energy of two (spin compensated) electrons in the same cloud: To a 1s cloud with Radius R we add the charge of a second 1s electron shell-wise. The potential energy of a spherical shell with area 4πr2 and thickness dr at radius r is:
We integrate from r=0 to r=R:
This result can also be found by letting two 1s clouds overlap until total fusion, see tech details.
For He we get:
hence σ = 0.4. The nuclear charge Z is screened by 0.4 charge units from the repulsion energy 3*0.4/R of the two electrons in the cloud.