When do angular momenta decouple again?

Whenever we have multiple basis it’s generally a good idea to look at the energy scales of different parts of the Hamiltonian. For example, consider the hydrogen atom Hamiltonian with the spin-orbit coupling: $$H=H_{coul} + H_{SO}$$ where $H_{coul}$ is the coulomb term and $H_{SO}$ is the spin-orbit coupling term. The energy scale of the coulomb term is of the order of $10$ eV and that of the spin-orbit coupling is of the order of $10^{-5}$ eV. What this means is that we can treat the spin-orbit as a perturbation to the coulomb term. So in this case, as the coulomb term commutes with individual z-component of angular momentum operators the $j_1,m_1;j_2,m_2$ basis is easier to work with.

With multielectron systems a third term comes into picture. The electron-electron correlation. And this term doesn’t commute with the individual $L_z$. So for these systems, if the correlation energy is much lower than spin-orbit (both are generally much lower than coulomb), then we can use the decoupled basis generally known as $ls$ basis. This is heavily used in spectroscopy, known as Russel-Saunders coupling. On the other hand if the correlation energy is higher than the spin-orbit coupling (as is the case in some actinides) we use the coupled basis generally known as $jj$ basis.