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 a first 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, for each of the three p-orbitals has two spin functions, so that in total there are 6 spinorbitals and 36 products. A check on dimensions before and after coupling is useful because it is easy to overlook coupled states.