The (n+1) Rule:
The (n+1) rule, also known as the Madelung rule or the diagonal rule, is a principle used to determine the order in which atomic orbitals are filled with electrons in an atom. It provides a guideline for filling electron shells and subshells.

Aufbau Principle:
The Aufbau principle is a fundamental principle in chemistry that states that electron orbitals are filled in order of increasing energy. According to this principle, electrons fill the lowest energy orbitals first before occupying higher energy levels. The Aufbau principle also follows the (n+1) rule, as it determines the order of electron filling based on the increasing values of n+ℓ.

Pauli's Principle:
Pauli's principle, formulated by Wolfgang Pauli, states that no two electrons in an atom can have the same set of four quantum numbers. The four quantum numbers include the principal quantum number (n), the azimuthal quantum number (ℓ), the magnetic quantum number (mℓ), and the spin quantum number (ms). Pauli's principle ensures that each orbital can accommodate a maximum of two electrons with opposite spins.

Hund's Principle:
Hund's principle, named after Friedrich Hund, states that for orbitals of equal energy, electrons occupy separate orbitals with parallel spins before pairing up. This principle is often referred to as "the bus seat rule" as it illustrates that electrons prefer to occupy empty orbitals before pairing up in the same orbital. Hund's principle allows for the maximization of electron-electron repulsion, which stabilizes the atom.

Application and Order of Filling:
The (n+1) rule, along with the Aufbau principle, Pauli's principle, and Hund's principle, guides the order of filling atomic orbitals. The order of filling is determined by the sum of the principal quantum number (n) and the azimuthal quantum number (ℓ). The filling order is as follows:

1. 1s
2. 2s
3. 2p
4. 3s
5. 3p
6. 4s
7. 3d
8. 4p
9. 5s
10. 4d
11. 5p
12. 6s
13. 4f
14. 5d
15. 6p
16. 7s
17. 5f
18. 6d
19. 7p

Conclusion:
The (n+1) rule, in conjunction with the Aufbau principle, Pauli's principle, and Hund's principle, provides a systematic approach to understanding the order of filling atomic orbitals with electrons. These principles play a crucial role in determining the electron configuration of atoms and predicting their chemical properties. By following these principles, scientists can accurately describe the distribution of electrons in atoms and explain various phenomena in chemistry.