The spherical harmonics represent the analytic solution to the Schrodinger equation for a hydrogen in a vacuum. Any other system (e.g. Hydrogen with two electro

The spherical harmonics represent the analytic solution to the Schrodinger equation for a hydrogen in a vacuum. Any other system (e.g. Hydrogen with two electrons, or a neutral Helium atom) does not have any analytical solution (its a many body problem). In general, determining the wave function for a collection of electrons is a very hard problem. Depending on the accuracy needed, different commonly used methods range from O(n^4) to O(n^7) where n is the number of electrons.