physical chemistry - While filling electrons, we follow Aufbau principle, but not while removing them. Why is this so?

I recently came across a question Why is the vanadium(3+) ion paramagnetic?, where the asker is wondering how $\ce{V^{3+}}$ is paramagnetic (he used Aufbau in reverse to remove the electrons), while the correct answerer to that question remarked that removing electrons must follow the order from outer shell to inner shell. For example, in the case of $\ce{V^{3+}}$ the electronic configuration of $\ce{V}$ is $\ce{[Ar] 3d^3 4s^2}$. The asker used reverse-Aufbau, which is $\ce{[Ar] 3d^0 4s^2}$; while the actuality was $\ce{[Ar] 3d^2 4s^0}$.

My question: Why does filling electrons follow a certain rule, while removing them follows a different rule?

Answer

Usually when adding electrons based on the Aufbau principle, you go from one element to the next highest one, e.g. from $\ce{Ti}: \ce{[Ar] 4s^2 3d^2}$ to $\ce{V: [Ar] 4s^2 3d^3}$. Thus you add not only an electron but also a proton to your atom.

When you remove electrons to get to a cation, you only remove electrons. Thus it is a different situation, with different interactions between nucleus and electrons, which affects the energy of the orbitals.

Also keep in mind, that the orbital picture is just a simplification. Strictly speaking (and from a purely theoretical point of view) orbitals do not exist, they are merely a mathematical crutch to solve the Schrödinger Equation. Although in practice this picture works surprisingly well, hence people use it a lot.

A proper quantum chemical calculation will give you the correct ground state, but you might have problems identifying it as something like $\ce{[Ar] 4s^2 3d^0}$. It will be a mixture (linear combination) of many electron configurations, although $\ce{[Ar] 4s^2 3d^0}$ might be the most important one for $\ce{V^{3+}}$.