Identical particles

Postulate: Every elementary particle is either a fermion or a boson.  A state of many identical particles is totally antisymmetric with respect to the interchange of any two particles if they are fermions, and it is totally symmetric if they are bosons.  No two identical fermions can have exactly the same set of quantum numbers.  This is called the Pauli exclusion principle.

Atomic Spectra

LS coupling
We assume that the non-central part of the electrostatic interaction is much bigger than the spin-orbit interaction.  (This is usually true for light multi-electron atoms.)  The electrostatic interaction leads to a splitting of the level corresponding to a given electron configuration into a number of sublevels characterized by different values of the total orbital angular momentum of the electrons, L, and their total spin, S.  The operator for the electrostatic interaction commutes with L = l₁+l₂+l₂+... and S = s₁+s₂+s₃+... .  Some of the values of L and S obtained from the general rules for addition of angular momenta can correspond to states forbidden by the Pauli principle.  Filled shells do not contribute to the total orbital angular momentum L and the total spin S.  To each allowed term LS belong (2L+1)(2S+1) states, differing by the values of M_L and M_S.  The spin-orbit interaction leads to a splitting of the term LS into a number of components corresponding to different values of the total angular momentum J.  But it does not completely remove the degeneracy.  Each J component is degenerate with a multiplicity of 2J+1.
∑_J(2J+1) = (2L+1)(2S+1).
In the LS coupling scheme, a term is designated by ^2S+1L_J.  2S+1 is called the multiplicity of the term.

Which terms corresponding to a given configuration have the lowest energy?

Hund's Rule (established empirically to find the ground state term)

• The level with the largest multiplicity has the lowest energy.
• For a given multiplicity, the level with the largest value of L has the lowest energy.
• For less than half-filled shells:
  • The component with the smallest value of J has the lowest energy (normal order).
    Examples of less than half-filled shells: np², nd²
• For more than half-filled shells:
  • The component with the largest value of J has the lowest energy (inverted order).
    Examples of more than half-filled shells: np⁴, nd⁸
• When the number of electrons is 2l + 1, i.e. when the shell is half filled, there is no multiplet splitting.