Question

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?

Answer 1

The procedure to pass from uncoupled to coupled basis set for total angular momenta in polyelectronic atoms involves evaluating the range of total L by exploiting the commutation between total Lz and individual momenta. The obtained wavefunctions are not completely exact due to the non-commutation of the Hamiltonian with the modulus of individual momenta. However, they can be considered as zero-th-order wavefunctions and can be improved by mixing them with combinations from other configurations.

Step-by-step explanation:

In the study of polyelectronic atoms, the formal procedure to pass from the uncoupled basis set of individual angular momenta to the coupled basis set of total angular momenta involves evaluating the range of total orbital angular momentum (L) by exploiting the commutation between total Lz and the individual angular momenta.

However, it is important to note that the Hamiltonian does not commute with the modulus of the individual momenta, which means that the wavefunctions obtained through this procedure are not completely exact.

These wavefunctions can be considered zero-th-order wavefunctions, as they provide a starting point for further improvements. One way to improve them is by mixing them with other combinations constructed in the same way and with the same type of terms (same L, same S), but from other electronic configurations.

This mixing can introduce off-diagonal elements in the Hamiltonian matrix, which corresponds to the interaction between different configurations.