Which orbital in a subshell gets filled first?

Question

Let's say we're filling electrons in subshell 2p. This subshell will have $m_l = -1$, $m_l = 0$, and $m_l = 1$ orbitals. Does the 'filling' of electrons depend on the value of $m_l$? Meaning, does the value of $m_l$ determine if it gets filled first by an electron or not?

Answer 1

The notation [-1,0,1] is given to address the sub shells.The energies of all these orbitals (unless you apply a magnetic field) are identical, therefore the filling is completely arbitrary. That means any of the shells can be filled first. But after 1 shell is filled the next electron goes to any one of the remaining shells. And the next goes to the other one. This is known as the Pauli's Exclusion law. Electrons with like spins are filled first and electrons with the opposite spin are filled next (note: 1 shell can share 2 electrons). This choice of spin is also complete arbitrary.

Answer 2

The only really important thing when filling electrons into lone atoms in vacuum is taking care to fill shells from lowest to highest principal quantum number $n$ and the azimuthal quantum number $l$, since those are the only two that affect the electron’s energy.

$n$ affects even the energies of electrons in hydrogen-like atoms while $l$ only comes into play once core electrons exist whose orbital shapes can make a difference to the overall charge distribution. This all makes macrophysical sense, since $n$ is directly part of the quantitised electron energy $E_n$ and $l$ is the quantum number determining the electron’s angular momentum with $l(l+1)\hbar$ being the eigenvalue of the squared angular momentum operator $\hat{\vec{L}^2}$.

The orientation of this (or more precisely: its $z$ component, $\hat{L_z}$ by its eigenvalue $m_l \hbar$) is determined by the magnetic quantum number your question is asking about. Stick to the image of a single atom in vacuum: There is no reason why certain directions of angular momentum (remember the value is the same, only the orientation changes) should in any way affect the energy — especially since $z$ is a completely arbitrary choice of coordinate; vacuum does not know up or down. It is not until you add surroundings, either by another atom approaching your atom in vacuum or by an external magnetic field that the orientation matters in any way.

So as long as you are adding electrons to single atoms aufbau-wise, you can choose whichever orbital you want to fill first. The energies of all three p-orbitals are identical and cannot be modified in any way. However, add another atom, and all of a sudden these become distinguishable. The system now starts disfavouring identical energy values of different orbitals if they aren’t completely populated, half-populated or unpopulated. This gives rise to effects such as Jahn-Teller distortion that you might encounter elsewhere. They all follow the basic principle of ‘reduce symmetry to distinguish directions to allow for different energy levels so my electrons can sit at lower levels’.