Rules for filling electrons in the Orbitals of an Atom

Since filling orbitals of an atom with electrons relies upon any of the following rules.

i) Aufbau Principle
Also known as the building up principle.
It's so called as it explains how electrons filling add up to build a configuration from lowest energy levels to those of highest in an atom.
According to this principle "The electrons in an atom are so arranged that they occupy orbitals in the order of their increasing energies."
Thus, the orbital with the lowest energy will be filled first, then the next i.e. higher in energy, and so on.
The reader should understand that, in the absence of any magnetic field, the energy of an orbital relies entirely on the principal quantum number (n) and the azimuthal quantum number (l), thus the order of filling orbitals with electrons may be obtained from the generalizations below.

a) The orbitals at which say n & L values are the lowest, usually is filled first.
E.g. 1s is filled before 2s.

b) If two orbitals have the same values of n & L, the orbital having the lower value of ml is filled first.
E.g. in 2p there are 3-orbitals filling starts with adding one electron into -1px then 0py to +1pz orbital.

Below is the order of filling electrons in orbitals by the Aufbau principle

1s, 2s. 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f,5d, 6p, 7s, 5f…

This is a simple way of working out this order.
As in this method a series of arrows running from upper right to the lower left gives the order of orbitals with increasing energy.

ii) Pauli’s Exclusion principle
Back in 1925 Wolfgang Pauli discovered what is known as the exclusion principle.
His principle was very useful in constructing the electronic configuration of atoms. According to his principle “No two electrons in an atom can have the same values for all the four quantum numbers”
Explained already if you landed on this page and doesn't know the meaning of the above principle please refer here for more.

For example in 1s orbital of helium atom there are two electrons.
According to the concept of quantum number and Pauli’s rule, their quantum numbers are shown below.

The (+) and (-) signs refers to the clockwise and anticlockwise spins of electrons.

Thus, the two electrons having the same values of n, L and ml must have different values of ms, i.e., their spins are in the opposite directions.

This leads to a very significant observation 

that "Each orbital can accommodate at the maximum only two electrons having opposite spins."

Implications of the Pauli's exclusion principles 

The Pauli's exclusion principle leads to the following conclusions:

(a) By no means an orbital can accommodate more than two electrons.

(b) In any main energy level (shell), the maximum number of electrons is twice the number of orbitals,

i.e. Number of electrons in any atom is given by 2n².

where; n is the principal quantum number.

iii) Hund's Rule of Maximum Multiplicity

The rules discussed above do not give any idea for filling electrons into the orbitals having equal energies (such states are called degenerate states).

For instance, three p-orbitals, i.e. px, py and

pz, have equal energies. There fore how should electrons be filled into these orbitals?

Think of an example into which three electrons are to be filled into three p-orbitals, these three electrons can be clearly filled into two different ways;

           px     py     pz

(I)

           px     py     pz

(II)

Now, which of the two is correct?

The answer to this question is given by Hund's rule,

As it states that,

"When more than one orbital of equal energies are present, then the electrons first occupy 

these orbitals separately with parallel spins. The pairing of electrons will only be after all the orbitals of a given sub-energy level are singly occupied."

As per Hund's rule, the correct way of filling three electrons in three p orbitals is that where each orbital is singly occupied, (arrangement II above).

Elaborations

Two electrons with parallel spins, tend to be as far apart as possible to minimize the electrostatic repulsion. Therefore, for this reason the electrons prefer more to occupy the orbitals singly first. As all the orbitals are singly occupied by the electrons, then the incoming electron is left with two choices to either pair up with the other electron or go after the next higher orbital. 

When vacant orbital of suitable energy is not available, then the incoming electron will have no choice but to pair up with that electron.

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