Spin-orbit coupling and atomic multiplets

In LS coupling, add individual electron orbital angular momenta to obtain $L$, and their spins to obtain $S$. Then couple these totals to $J$. This order is useful when electrostatic term separations dominate spin-orbit mixing. The term symbol is $^{2S+1}L_J$, using capital S, P, D, and F for orbital values zero through three. The letter S and the spin symbol S play different roles, so read the superscript and the letter separately.

For specified $L$ and $S$, the allowed $J$ values run in unit steps from $|L-S|$ through $L+S$. Each level has $2J+1$ magnetic substates in zero external field. Summing their counts reproduces $(2L+1)(2S+1)$. This is a useful completeness check when nonequivalent angular momenta are coupled. For equivalent electrons, antisymmetry can forbid whole LS terms before this coupling step; naive addition of distinguishable spins and orbitals would overcount the physical states.

Within an isolated LS term, an effective spin-orbit Hamiltonian may be written $H_{\mathrm{so}}=A\mathbf L\cdot\mathbf S/\hbar^2$, where $A$ has energy units. Squaring $\mathbf J=\mathbf L+\mathbf S$ expresses the dot product through three commuting squared angular momenta. This eliminates any need to assign definite classical directions to L and S. Their magnitudes are quantum eigenvalues, and a coupled state generally lacks separate definite projections for the two vectors.

The resulting level shifts have a weighted center of gravity at zero relative to the unsplit term. Adjacent levels satisfy the Landé interval rule: the difference between levels J and J minus one is $AJ$. The sign of A decides their ordering. That sign is not fixed universally by the notation; for a many-electron term it can reflect shell filling and effective interactions. If observed intervals strongly violate the rule, configuration interaction or intermediate coupling may be physically relevant.

In a rotationally invariant atom, total J and its projection remain valuable labels even when LS coupling fails. For heavier atoms, individual electron spin-orbit energies can compete with electrostatic separations. A jj-coupled starting basis then couples each electron orbital and spin first, followed by the individual j values. Neither basis changes the Hilbert space. The quality of the approximation depends on which Hamiltonian terms are diagonal or small in that basis.

A multiplet in an observed spectrum contains transitions between several levels, not merely the levels of one term drawn horizontally. Different allowed transitions can share or nearly share frequencies, and their intensities require matrix elements and populations. Counting substates therefore does not predict an intensity pattern by itself. In the two-state lab, phase changes illustrate why probabilities in one basis fail to specify a state. It has no controls for L, S, exchange integrals, or a real multiplet Hamiltonian.