quantum mechanics - How can we describe the electrons of multi-electron atoms (i.e. not Hydrogen) when equations/analytic solutions only exist for Hydrogen?

I've been digging into emission spectra of different elements and found that such things as the Rydberg equation, Bohr's model, and quantum mechanics can only fully describe the single electron in the Hydrogen atom. How did we then make the leap to s,p,d,f shells of multi-electron atoms? How accurate is our analysis of these more complicated elements?

Rydberg Equation (side-note: Is this an empirical 'data-fitting' equation? What's the significance of that?)

$$\frac{1}{\lambda}=R_H\left( \frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$

Hydrogen:

Helium:

Iron:

Potassium:

Answer

The only atoms for which the Schrodinger equation has an analytic solution are the one electron atoms i.e. H, He$^+$, Li$^{2+}$ and so on. That's because with more than one electron the forces between electrons make the equation too hard to solve analytically. However, over the 90 or so years since Schrodinger proposed his equation a vast array of numerical methods for solving it have been developed, and of course modern computers are so powerful they can calculate the (electronic) structure of any atom with ease. This applies even to heavy atoms where relativistic effects need to be taken into account.

The Rydberg equation is an approximation because it does not take the electronic fine structure into account. However it's a pretty good approximation. It works because for a one electron atom the energy of the orbitals (ignoring fine structure) is proportional to 1/$n^2$, where $n = 1$ is the lowest energy orbital, $n = 2$ is the second lowest and so on.