Give the electron configuration of (a) \(\mathrm{Pm} ;\) (b) \(\mathrm{Lu}^{3+} ;\) (c) \(\mathrm{Th}\) (d) \(\mathrm{Fm}^{3+}\).

The Aufbau principle is a fundamental rule for determining the electron configuration of atoms.
According to this principle, electrons occupy the lowest energy orbitals available first.
In simpler terms, orbitals are filled sequentially from the lowest energy level to the highest.
This helps explain why electrons in an atom fill specific energy levels before moving on to higher ones.

For example, in step 3 of the exercise, we made use of the Aufbau principle to organize the 61 electrons of Promethium.
The sequence started with the lowest energy orbitals and moved to higher ones: ewline ewline 1s^2 → 2s^2 → 2p^6 → 3s^2 → 3p^6 → 4s^2 → 3d^{10} → 4p^6 → 5s^2 → 4d^{10} → 5p^6 → 6s^2 → 4f^5.

This systematic arrangement ensures that the electron distribution minimizes the atom's energy, thereby achieving a more stable configuration.

The Pauli exclusion principle is another essential rule for understanding electron configuration.
This principle states that no two electrons in an atom can have the same four quantum numbers.
In simpler words, an atomic orbital can hold a maximum of two electrons, and they must have opposite spins.

This principle prevents electrons from crowding into the same energy state and creates a system where electrons are distributed in various orbitals and energy levels.

When we wrote the electron configuration for elements like Thorium (Th) in step 5, we made sure that within each orbital, if an electron pair exists, they have opposite spins. ewline ewline For instance, in the 2p^6 subshell, six electrons are distributed as follows: the first three fill the three p orbitals with same spins (↑), and the next three fill the same orbitals but with opposite spins (↓). This efficient filling keeps all electrons in their respective energy states as per the principle.

Hund's rule is crucial for understanding how electrons distribute themselves within a subshell.
According to Hund's rule, electrons will fill degenerate orbitals (orbitals of the same energy level) singly before pairing up.
Electrons prefer to be unpaired because this minimizes electron-electron repulsion and results in a lower energy configuration.

For instance, in our step 6 while distributing electrons for the Fm^3+ ion (Fermium ion), we applied Hund's rule for the 5f subshell.
There are 11 electrons in the 5f orbitals: ewline ewline Instead of filling one f orbital completely before moving to the next, all seven 5f orbitals would first get one electron each (↑), and then the remaining four orbitals would get a second electron (↓), following Hund's rule.

This results in a configuration that is more stable as it minimizes repulsion between the electrons within the same subshell by maximizing the number of unpaired electrons.