In this video https://www.youtube.com/watch?v=tq_y1qOmUBE&t=783s it's mentioned that the structor of atoms of multiple particles can be approximated using the Schrodinger Equation of the Hydrogen Atom. The number that can fit in each shell is the number of configurations for the corresponding energy level for the hydrogen atom multiplied by 2.
I know that for the case of the multi electron atom $$V_n=-\frac{e^2}{4\pi\varepsilon_0r_n}$$ and $$V_{mn}=\frac{e^2}{4\pi\varepsilon_0r_{mn}}$$
Let's say that instead we have a bound state, for which $$V_n=f(r_n)$$ and $$V_{mn}=-f(r_{mn})$$ with $$V(r)\not\propto-\frac{1}{r}$$
and each particle in this bound state is an identical fermion.
Can we approximate the structor of this system using the Time Independent Schrodinger equation for the case of a single particle with $V=f(r)$?