Hund's Rule (Quantum Numbers)

name	symbol	orbital meaning	range of values	value example
principal quantum number	n	shell	1 ≤ n	n = 1, 2, 3, …
azimuthal quantum number (angular momentum)	ℓ	subshell (s orbital is listed as 0, p orbital as 1 etc.)	0 ≤ ℓ ≤ n − 1	for n = 3: ℓ = 0, 1, 2 (s, p, d)
magnetic quantum number, (projection of angular momentum)	mℓ	energy shift (orientation of the subshell's shape)	−ℓ ≤ mℓ ≤ ℓ	for ℓ = 2: mℓ = −2, −1, 0, 1, 2
spin projection quantum number	ms	spin of the electron (−½ = counter-clockwise, ½ = clockwise)	−½, ½	for an electron, either: −½, ½

This model describes electrons using four quantum numbers, n, ℓ, m_ℓ, m_s. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons).

• The first, n, describes the electron shell, or energy level.
  • The value of n ranges from 1 to "n", where "n" is the shell containing the outermost electron of that atom. For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. This is known as the principal quantum number.
• The second, ℓ, describes the subshell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).
  • The value of ℓ ranges from 0 to n − 1. This is because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on. A quantum number beginning in 3, 0, … describes an electron in the s orbital of the third electron shell of an atom.
• The third, m_ℓ, describes the specific orbital (or "cloud") within that subshell.*
  • The values of m_ℓ range from −ℓ to ℓ. The s subshell (ℓ = 0) contains only one orbital, and therefore the m_ℓ of an electron in an s subshell will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the m_ℓ of an electron in a p subshell will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with m_ℓ values of −2, −1, 0, 1, and 2.
• The fourth, m_s, describes the spin of the electron within that orbital.*
  • An electron can have a spin of ±½, m_s will be either, corresponding with "spin" and "opposite spin." Each electron in any individual orbital must have different spins, therefore, an orbital never contains more than two electrons.

* Note that, since atoms and electrons are in a state of constant motion, there is no universal fixed value for m_ℓ and m_s values. Therefore, the m_ℓ and m_s values are defined somewhat arbitrarily. The only requirement is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p subshell could be described as m_ℓ= −1 or m_ℓ = 0, or m_ℓ = 1, but the m_ℓ value of the other electron in that orbital must be the same, and the m_ℓ assigned to electrons in other orbitals must be different).