Energies of Orbitals

In hydrogen and in hydrogen-like species (one-electron systems), the ONLY interaction present is the attraction between the single electron and the nucleus, so orbital energy depends purely on the principal quantum number n; orbitals sharing an n value but differing in shape — such as 2s and 2p — are exactly equal in energy, i.e. degenerate, giving the ordering $1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < \ldots$. In a MULTI-electron atom this simple picture breaks down, because the electrons now repel one another as well as being attracted to the nucleus, so orbital energy depends on BOTH n and the azimuthal quantum number l. The rule that governs this is the $(n+l)$ rule: the lower the sum $(n+l)$ for a given orbital, the lower its energy; and if two orbitals happen to tie on $(n+l)$, the one with the SMALLER n is the lower-energy orbital (for example, 2p and 3s both have $n+l=3$, but 2p, with the smaller n, sits at lower energy than 3s). Applying the $(n+l)$ rule systematically across all the orbitals gives the standard multi-electron filling order $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s\ldots$, which is exactly the order electrons fill as atoms are built up (see the Aufbau princip …