What causes different energy levels in atoms with multiple electrons?

[Goodver]

From the calculations of total energy on an energy level we end up with rhe equation which consists of constants only, except of the principal quantum number.

Since equation does not include any unique variables, such as number of protons or electrons in an atom, what makes then different atoms have diferent energy levels?

these calculations for Hydrogen, I assume, the number of electrons somehow determines the energies on the levels, so for different atom with more than 1 electron equations should vary. I also assume that reduced mass caused by different number of protons for different atoms should not cause much of influence.

or energy on a level is a sort of a superposition of waves of all electrons on this level? => sum of energies?

what determines unique energies on levels for different atoms?

 

[DrClaude]

These equations are derived for a nucleus consisting of a single proton, with one electron "orbiting" around it. Very similar equations can be obtained for any charge $Z$ of the nucleus, but again with only one electron. If more than one electron is present, then there are addition terms in the Hamiltonian corresponding to the Coulomb interaction between electrons. This breaks the spherical symmetry and such simple equations can't be derived anymore. Only in the special case of alkali atoms (single valence electron), can an approximate simple formula for the energy of the valence electron be obtained (quantum defect theory).