Russell Saunders coupling

Russell Saunders coupling, or L-S coupling, is a method to derive the total angular momentum of a light multi-electron atom from its component angular momenta. It assumes that spin-spin coupling > orbit-orbit coupling > spin-orbit coupling. This coupling scheme leads to the grouping of degenerate eigenstates into terms and levels, both of which correspond well with spectroscopic data of elements in the first three periods of the periodic table.

The total angular momentum $\boldsymbol{\mathit{j}}$ of a 1-electron atom is defined as the vector sum $\boldsymbol{\mathit{j}}=\boldsymbol{\mathit{l}}+\boldsymbol{\mathit{s}}$ , where $\boldsymbol{\mathit{l}}$ and $\boldsymbol{\mathit{s}}$ are the orbital angular momentum and spin angular momentum respectively of the electron in the atom. For an $n$ -electron atom, Henry Russell and Frederick Saunders proposed that the total angular momentum $\boldsymbol{\mathit{J}}$ of the atom is $\boldsymbol{\mathit{J}}=\boldsymbol{\mathit{L}}+\boldsymbol{\mathit{S}}$ , where $\boldsymbol{\mathit{L}}=\sum_{i=1}^n\boldsymbol{\mathit{l}}_i$ and $\boldsymbol{\mathit{S}}=\sum_{i=1}^n\boldsymbol{\mathit{s}}_i$ . If we use a 2-electron atom as an example, the corresponding operator is:

$\hat{J}=\hat{L}\otimes I+I\otimes\hat{S}\; \; \; \; \; \; \; \; 290$

Since spin-spin coupling > orbit-orbit coupling > spin-orbit coupling, the spin angular momenta couple amongst themselves and the orbital angular momenta couple amongst themselves, before the total spin angular momentum couples with the total angular momentum. Hence, we have $\hat{L}=\hat{l}^{(1)}\otimes I+I\otimes\hat{l}^{(2)}$ and $\hat{S}=\hat{s}^{(1)}\otimes I+I\otimes\hat{s}^{(2)}$ . The eigenstates of $\hat{L}$ are coupled representations of the basis vectors of $\hat{l}^{(1)}$ and $\hat{l}^{(2)}$ , while the eigenstates of $\hat{S}$ are coupled representations of the basis vectors of $\hat{s}^{(1)}$ and $\hat{s}^{(2)}$ . The eigenstates of $\hat{J}$ can either have the form of an uncoupled representation $\vert L,M_L,S,M_S\rangle$ or a coupled representation $\vert J,M_J,L,S\rangle$ . The uncoupled representation is used when spin-orbit coupling is weak.