LS coupling ($\mathbf{J}=\mathbf{L}+\mathbf{S}$)

An atomic addition of angular momentum scheme in which the individual orbital angular momentum vectors first combine into $\mathbf{L}$ and the individual spins into $\mathbf{S}$, then these combine into total angular momentum $\mathbf{J}$. When spin-orbit coupling is weak compared with the interactions between electrons described by electrostatics, $L$ and $S$ remain useful approximate quantum numbers. The Landé g-factor describes the resulting weak-field magnetic splitting.

For electrons in distinct radial orbitals, angular coupling permits all $L$ from $|l_1-l_2|$ to $l_1+l_2$ and both total spins $S=0,1$. A p-s pair therefore gives singlet and triplet P terms. A nonequivalent p-p pair gives singlet and triplet S, P and D terms. The equivalent-subshell exclusion restrictions do not remove these nonequivalent terms. Counting $(2S+1)(2L+1)$ gives 12 p-s and 36 p-p microstates, matching products of one-electron state counts.

Two equivalent $l=1$ electrons require an antisymmetric complete state. Orbital exchange has parity $(-1{)}^L$ and the spin singlet is antisymmetric, giving allowed $L+S$ even. The terms are ${}^3{P}_{0,1,2}$, ${}^1{D}_2$ and ${}^1{S}_0$, with $9+5+1=15$ states, equal to $\left(\genfrac{}{}{0ex}{}{6}{2}\right)$. Hund ordering puts ${}^3{P}_0$ lowest for the less-than-half-filled configuration.