I've studied the formal procedure to pass from the uncoupled basis set of individual angular momenta to the coupled basis set of total angular momenta for polyelectronic atoms.
I start from an electronic configuration, write down all the possible combinations of quantum numbers, each row is a Slater determinant, and then evaluate the range of total L exploiting the commutation between total Lz and the individual ones. I can apply ladder operators to find the "correct" linear combinations of Slater determinant (rows) which determine eigenfuntions of the CSCO of the coupled set. Pretty clear so far.
Nevertheless i have read that Hamiltonian doesn't commute with the modulus of individual momenta. But this procedure mixes microstates with the same individual quantum numbers l (same configuration). Is this the reason why these wavefunctions are not completely exact? However, can we say they're correct zero-th order wave function and why?
Can I improve them by mixing with combinations constructed in the same way and with the same type of term (same L, same S) but from others configurations? So, is the Hamiltonian matrix diagonal in this momenta CSCO only limiting to a certain configuration and off-diagonal elements appears when extended to more configurations?
