 Landé interval rule

For dimensionless angular momentum operators and a Hamiltonian $H=a\mathbf{I}\cdot \mathbf{J}$ at fixed $I,J$, addition of angular momentum gives $E_F=a[F(F+1)-I(I+1)-J(J+1)]/2$. Subtracting neighbouring energy levels proves the displayed rule. For fine structure substitute $\mathbf{L},\mathbf{S}$ and total $J$; for magnetic-dipole hyperfine structure use electronic $J$, nuclear spin $I$ and total $F$. Electric-quadrupole interactions and mixing with nearby energy levels can violate the rule.

The Landé interval rule is a principle used in atomic and molecular physics, specifically in the context of spectroscopic transitions and the determination of energy levels. It provides a way to estimate the intervals between energy levels of complex systems, like atoms with multiple electrons, which can be challenging to analyze due to electron-electron interactions and the configuration of electrons. The rule helps predict the positions of spectral lines in atomic spectra by considering the contributions of different electron configurations to the total angular momentum of the system.