Atomic Spectroscopy  Allowed Terms
8. Allowed Terms of Levels for Equivalent Electrons

LS Coupling
The allowed LS terms of a configuration consisting of two nonequivalent groups of electrons are obtained by coupling the S and L vectors of the groups in all possible ways, and the procedure may be extended to any number of such groups. Thus the allowed terms for any configuration can be obtained from a table of the allowed terms for groups of equivalent electrons.
The configuration l^{ N} has more than one allowed term of certain LS types if l > 1 and 2 < N < 4l (d^{ 3}  d^{ 7}, f^{ 3}  f^{ 11}, etc.). The recurring terms of a particular LS term type from d^{ N} and f^{ N} configurations are assigned sequential index numbers in the tables of Nielson and Koster [7]; the index numbers stand for additional numbers having grouptheoretical significance that serve to differentiate the recurring terms, except for a few terms of f^{ 5} and f^{ 9}, f^{ 6} and f^{ 8}, and f^{ 7}. These remaining terms, which occur only in pairs, are further labeled A or B to indicate Racah's separation of the two terms.
The index numbers of Nielson and Koster are in practice the most frequently used labels for the recurring terms of f^{ N} configurations. Use of their index numbers for the recurring terms of d^{ N} configurations has perhaps the disadvantage of substituting an arbitrary number for a quantum number (the seniority) that itself distinguishes the recurring terms in all cases. The actual value of the seniority number is rarely needed, however, and a consistent notation for the d^{ N} and f^{ N} configurations is desirable. A table of the allowed LS terms of the l^{ N} electrons for l ≤ 3 is given in Ref. [8], with all recurring terms having the index numbers of Nielson and Koster as a following online integer. The theoretical group labels are also listed. Thus the d^{ 3} ^{2}D term having seniority 3 is designated ^{2}D2, instead of ^{2}_{3}D, in this scheme; and the level having J = ^{3}/_{2} is designated ^{2}D2_{3/2}.

jj Coupling
The allowed J values for a group of N equivalent electrons having the same j value, l_{j}^{N}, are given in the table below for j = ^{1}/_{2}, ^{3}/_{2}, ^{5}/_{2}, and ^{7}/_{2} (sufficient for l ≤ 3). The l^{ 4}_{7/2} group has two allowed levels for each of the J values 2 and 4. The subscripts distinguishing the two levels in each case are the seniority numbers [9].
Allowed J values for l_{j}^{N} equivalent electrons (jj coupling).
l_{j}^{N} 
Allowed J values 

l_{1/2} 
^{1}/_{2} 
l^{2}_{1/2} 
0 
l_{3/2} and l_{3/2}^{3} 
^{3}/_{2} 
l_{3/2}^{2} 
0, 2 
l_{3/2}^{4} 
0 
l_{5/2} and l_{5/2}^{5}

^{5}/_{2} 
l_{5/2}^{2} and l_{5/2}^{4} 
0, 2, 4 
l_{5/2}^{3} 
^{3}/_{2}, ^{5}/_{2}, ^{9}/_{2} 
l_{5/2}^{6} 
0 
l_{7/2} and l_{7/2}^{7}

^{7}/_{2} 
l_{7/2}^{2} and l_{7/2}^{6} 
0, 2, 4, 6 
l_{7/2}^{3} and l_{7/2}^{5} 
^{3}/_{2}, ^{5}/_{2}, ^{7}/_{2}, ^{9}/_{2}, ^{11}/_{2}, ^{15}/_{2} 
l_{7/2}^{4} 
0, _{2}2, _{4}2, _{2}4, _{4}4, 5, 6, 8 
l_{7/2}^{8} 
0 
The allowed levels of the configuration nl^{ N} may be obtained by dividing the electrons into sets of two groups $$nl^Q_{l+1/2} ~ nl^R_{l1/2} ~,$$ Q + R = N. The possible sets run from Q = N  2l (or zero if N > 2l) up to Q = N or Q = 2l + 2, whichever is smaller. The (degenerate) levels for a set with both Q and R nonzero have wave functions defined by the quantum numbers (αJ_{1}, βJ_{2}) J, with J_{1} and J_{2} deriving from the Q and R groups, respectively. The symbols α and β represent any additional quantum numbers required to identify levels. The J values of the allowed levels for each (αJ_{1}, βJ_{2}) subset are obtained by combining J_{1} and J_{2} in the usual way.