Russell-Saunders coupling

In atomic spectroscopy, Russell-Saunders coupling, also known as L-S coupling, specifies the coupling of electronic spin- and orbital-angular momenta. In the Russell-Saunders coupling scheme—called after H. N. Russell and F. A. Saunders—the orbital angular momenta of the electrons are coupled to total angular momentum with quantum number L, and the spin angular momenta are coupled to total S. The resulting L-S eigenstates are characterized by term symbols.

As an example we consider the excited helium atom in the atomic electron configuration 2p3p. B the triangular conditions the one-electron spins s = ½ can couple to |½−½|,  ½+½ = 0,  1 (spin singlet and triplet) and the two orbital angular momenta l = 1 can couple to L = |1−1|, 1, 1+1 = 0, 1, 2. In total, Russell-Saunders coupling gives two-electron states labeled by the term symbols:

¹S, ¹P, ¹D, ³S, ³P, ³D,

The dimension is 1×(1+3+5) + 3×(1+3+5) = 36. The electronic configuration 2p3p stands for 6×6 = 36 orbital products, as each of the three p-orbitals has two spin functions, so that in total there are 6 spinorbitals. A check on dimensions before and after coupling is useful because it is easy to overlook coupled states.

Russell-Saunders coupling gives useful first-order states in the case that one-electron spin-orbit coupling is much less important than the Coulomb interactions between the electrons. This occurs for the first part of the periodic table, roughly up to Z = 80. The usefulness stems from the fact that states of different L and S do not mix under the total Coulomb interaction.

In the lower regions of the periodic system it is more useful to first couple the one-electron momenta j ≡ l + s and then the one-electron j-eigenstates to total J. This so-called j-j coupling scheme gives a better first-order approximation when spin-orbit interaction is larger than the Coulomb interaction. If, however, in either coupling all resulting states are accounted for, the same subspace of Hilbert (function) space is obtained and the choice of coupling scheme is irrelevant.

More complicated electron configurations

(To be continued)