Russell-Saunders coupling

In atomic spectroscopy, Russell–Saunders coupling, also known as L–S coupling, specifies a coupling scheme of electronic spin- and orbital-angular momenta.

In Russell-Saunders coupling—called after H. N. Russell and F. A. Saunders—the orbital angular momentum eigenstates of the electrons are coupled to eigenstates with quantum number L of the total angular momentum operator L. Separately the one-electron spin functions are coupled to eigenstates with quantum number S of total S. Sometimes there is further coupling to J ≡ L + 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. By 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 with principal quantum number n = 2 and also 6 spinorbitals with n = 3. 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 and can be neglected. This occurs for the higher part of the periodic table, roughly down to Z = 80. The usefulness stems from the fact that states of different L and S do not mix under the total Coulomb interaction, so that L-S coupling achieves a considerable block diagonalization of the matrix of a Hamiltonian in which spin-orbit coupling is absent.

In the lower regions of the periodic system it is common 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 more useful first-order approximation when spin-orbit interaction is larger than the Coulomb interaction and spin-orbit interaction is included, while the Coulomb interaction is neglected. If, however, in either coupling scheme all resulting states are accounted for, i.e., the same subspace of Hilbert (function) space is obtained, then the choice of coupling scheme is irrelevant in calculations where both interactions—electrostatic and spin-orbit—are included on equal footing.

More complicated electron configurations

(To be continued)