# LS-coupling, residual electrostatic interaction and interval rule

I have a little bit of a confusion about the terminology. Are LS-coupling and residual electrostatic interaction the same thing? Or is it more correct to say that LS coupling is an approximation made to treat residual electrostatic interaction as a perturbation under conditions that spin-orbit coupling is weak? Can be LS-coupling still be used when spin-orbit coupling is not weak? And how is Lande interval rule derived for LS-coupling?

• I would tend to associate LS coupling as a spin-orbit coupling in atoms where it isn't too strong. What literature makes you think otherwise? Commented May 13, 2017 at 12:38
• See also physics.stackexchange.com/questions/80483/… . I think that, for a single-electron system, "LS coupling" is a synonym of "spin-orbit coupling", though I could be wrong. Commented May 16, 2017 at 23:33

Okay, I think I've managed to figure out this myself in the end. At a very simplistic level of my understanding, when one talks about LS-coupling one means the fact that in atom spins and orbital momenta $\mathbf{s}_i$ and $\mathbf{l}_i$ of the individual electrons are not constants of motion and thus are not good quantum numbers due to coulomb interactions and spin-spin interactions of the electrons. However, due to the conservation of angular momentum, their respective total $\mathbf{S}$ and $\mathbf{L}$ are still constants of motion.
Meanwhile, when one talks about SO-coupling one means the effect of the interaction of spin of the electron with its orbital motion. One can see it as the electron experiencing the electric field of the atom as a B-field in its rest frame. This leads to the interval rule if one does the proper derivation of the resultant energy shift. However, if SO-coupling is weaker than the effects of LS-coupling, one can no longer use the $\mathbf{s}_i$ and $\mathbf{l}_i$ for each individual electron. But $\mathbf{S}$ and $\mathbf{L}$ are still valid quantum numbers. That means that SO-coupling results in the precession of $\mathbf{S}$ and $\mathbf{L}$ around their total $\mathbf{J}$. Thus, projections of $\mathbf{S}$ and $\mathbf{L}$ on to $\mathbf{J}$ remain constants of motion, so the SO-coupling can be calculated by doing the same derivation, but using $\mathbf{S}$ and $\mathbf{L}$ this time, giving the interval rule in terms of $J,S,L$. Feel free to correct me if I am wrong!