How is the central potential calculated in the central field approximation?

In the central field approximation for atoms, one first guesses a central potential and then solves the TISE to obtain the electron wavefunctions. The self-consistent field approach now calculates an improved central potential using the wavefunctions and iterates the process.

My question is, how is the improved potential actually found using the wavefunctions? My guess would be to calculate the charge distribution due to all the electrons, find the potential due to that, and then average it over the solid angle to make it central. The problem with this is that a given electron now feels the potential due to itself, which is surely not very accurate, so I am wondering if there is a better way.