Energy of electrons in the same subshell

Question

So I have this quote from wikia (it's similar to my lecture pdf file) but I don't fully understand the statement

Although it is commonly stated that all the electrons in a shell have the same energy, this is an approximation. However, the electrons in one subshell do have exactly the same level of energy,[6] with later subshells having more energy per electron than earlier ones. This effect is great enough that the energy ranges associated with shells can overlap

In this statement, does it mean for example:

1) "Each" electron in the subshell 1s and "each" electron in subshell 2s have the same amount of kinetic energy.

Or

2) It means :"All electrons 1s have the amount of Kinetic Energy & All electrons 2s have the same amount of Kinetic Energy. However each electron in 1s doesn't have the same amount of Kinetic Energy compared to one electron in 2s" ?

Answer 1

It's important to remember (or to learn, if it hasn't been explained before) that the only atomic systems for which we can resolve the Schrödiger equation exactly are what we call hydrogenoid atoms - atoms with only one electron, such as $\ce{H}$, $\ce{He+}$, $\ce{Li^2+}$...

As soon as you add a second electron, atomic orbitals change and they aren't exactly the same as for the hydrogen atom (this is a form of the three-body problem that also comes up in other fields of physics; if you remember your Newtonian mechanics, that's why you cannot solve analytically a three-body gravitational problem). The source of this change is electron-electron interaction.

However, electron-electron interaction, in general, introduces only minor distortions in atomic orbitals - for the most part, the orbitals of poly-electronic atoms are very very similar to hydrogenoid atoms in terms of shape, size, etc, which is why we use the same labels for them as for hydrogenoid atoms ($\mathrm{1s}$, $\mathrm{2s}$, $\mathrm{2p}$, etc).

There is one property of hydrogenoid atoms that is very importantly changed by electron-electron interactions, though: in hydrogenoid atoms, the energy of an orbital is only a function of its main quantum number, $\mathrm{n}$, which means that all the orbitals of the same shell will have the same energy. In a single-electron atom, an electron in a $\mathrm{2s}$ orbital will have exactly the same energy as an electron in a $\mathrm{2p}$ orbital, because they only interact with the nucleus through a Coulombic interaction - which has spherical symmetry (it's the same in all the directions of space; it only changes with distance to the nucleus).

If there are other electrons in the same atom, occupying different orbitals, this breaks down the energy equivalence of same-shell orbitals - because, for instance, an electron in a $\mathrm{1s}$ orbital interacts differently with an electron in a $\mathrm{2s}$ orbital or in a $\mathrm{2p}$ orbital - due to symmetry, overlap, and a number of factors (these interactions are very complex, can't be solved analytically, and to some extent aren't even completely understood). The energy corrections are generally small, but the orbitals belonging to the same shell are no longer equal in energy - that's why we can order subshells, and we know that $\mathrm{2s}$ orbitals fill up before $\mathrm{2p}$ orbitals.

Answer 2

Let's be clear. The electrons in the same sub-shell have the same energy when not in an external electromagnetic field. Since electromagnetic fields are infinite in size, it is only an approximation to claim that same shell electrons all have identical energy. (It can be a good approximation, but it is an approximation nevertheless). For the same type of atom (same nuclear charge), the energy of all the electrons in the same sub-shell are (approx.) the same. I'm not sure how you jumped from that to thinking that all s or p or d or f or g or h or.... sub-shells were of the same energy. Of the captured electrons quantum numbers, the Principle quantum number affects energy the most. Keep in mind that the sub-shells are not necessarily relevant. For instance, hydrogen has a 1f subshell and it is possible that an excited H atom could have an electron in that subshell. But normally we speak about ground state configuration. It should come as no surprise if I were to tell you that the 1f subshell required more energy than, say the 2s subshell of a hydrogen atom. (Although I do not know which is actually higher energy, I'd have to either do the calculation or check the tables). So, it seems to me that the question isn't whether or not subshells with lower Principle Quantum Number can be higher energy than subshells with higher Principle QN, rather the question is do occupied (in the ground state) subshells ever show that property. The Aufbau Principle predicts the 4s is filled before the 3d, etc. https://en.wikipedia.org/wiki/Aufbau_principle which clearly answers that question. (Since aufbau says the lower energy level is filled first). (Note that there are well known exceptions to the rule, take a look at the ground state electron occupancies of Cr or Cu, they are not what you'd expect from a naive application of aufbau (and it gets worse as atomic number increases, "worse" in the sense of more exceptions)). To be clear, a Nx subshell electron will never have more energy than a (N+1)x subshell electron* (except possibly during the transition from one energy level to another, but thinking of an electron which has absorbed more energy than it can have in a sub-shell, thinking of that electron as still in that sub-shell, is a matter of definition and I wouldn't buy it) *Where N and N+1 are the Principle quantum no. and x from the set (spdfgh....). I'll also add the fact that only the s subshells are spherically symmetrical, so all of the other shells will be oriented with respect to any external electromagnetic field, hence various sub-subshells will have different energies because they're differently oriented towards that field. I should for completeness also note that two electrons share the same sub-sub-shell only if they have opposite spins, (which again will cause slightly different energies if in an external field).