# Energy levels hydrogenic atoms

#### Chemist20

I don't get why for hydrogenic atoms the 2s and 2p orbitals have the same energy. i do get it mathematically, but im thinking that the fact that there are angular nodes in 2p and not in 2s MUST affect the energy!!!!

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#### Bill_K

It's an "accidental" degeneracy of course, but qualitatively here's why: the energy depends only on the principal quantum number n = ℓ + nr + 1 where nr is the radial quantum number, i.e. the number of radial nodes. And so more nodes in the angular direction tends to increase the energy, but it is accompanied by fewer nodes in the radial direction which tends to decrease it.

#### Chemist20

It's an "accidental" degeneracy of course, but qualitatively here's why: the energy depends only on the principal quantum number n = ℓ + nr + 1 where nr is the radial quantum number, i.e. the number of radial nodes. And so more nodes in the angular direction tends to increase the energy, but it is accompanied by fewer nodes in the radial direction which tends to decrease it.

Uhm okay, but I still don't get why for the 2s being more core-like than 2p, for the hydrogenic atom this isn't taken into account and hence both orbitals have the same energy.

why does it only take it into account when we talk about multi electron atoms??

#### jtbell

Mentor
It's an "accidental" degeneracy of course, but qualitatively here's why: the energy depends only on the principal quantum number n = ℓ + nr + 1 where nr is the radial quantum number
This holds only for the Coulomb potential, V ≈ -1/r, IIRC.

why does it only take it into account when we talk about multi electron atoms??
In a multi-electron atom, an individual electron "feels" not only the attraction of the nucleus,but also the repulsion of the other electrons. The "effective" potential is not -1/r as with a one-electron atom.

#### phyzguy

I don't get why for hydrogenic atoms the 2s and 2p orbitals have the same energy. i do get it mathematically, but im thinking that the fact that there are angular nodes in 2p and not in 2s MUST affect the energy!!!!
Your intuition is correct. In reality they don't have the same energy. The relativistic corrections and the spin-orbit coupling breaks the degeneracy.

#### Chemist20

Your intuition is correct. In reality they don't have the same energy. The relativistic corrections and the spin-orbit coupling breaks the degeneracy.
Right... so then why in the H atom the 2s has the same energy as the 2p ?

also... how does spin coupling affect the energies. as far as I knew, spin coupling arises due to the interaction of the orbital angular momentum and spin angular momentum. So....?