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 ; the index numbers stand for additional numbers having group-theoretical 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. , with all recurring terms having the index numbers of Nielson and Koster as a following on-line integer. The theoretical group labels are also listed. Thus the d 3 2D term having seniority 3 is designated 2D2, instead of D, in this scheme; and the level having J = 3/2 is designated 2D3/22.
|ljN||Allowed J values|
|l3/2 and l 33/2||3/2|
|l 23/2||0, 2|
|l5/2 and l 55/2||5/2|
|l 25/2 and l 45/2||0, 2, 4|
|l 35/2||3/2, 5/2, 9/2|
|l7/2 and l 77/2||7/2|
|l 27/2 and l 67/2||0, 2, 4, 6|
|l 37/2 and l 57/2||3/2, 5/2, 7/2, 9/2, 11/2, 15/2|
|l 47/2||0, 22, 42, 24, 44, 5, 6, 8|
The allowed levels of the configuration nl N may be obtained by dividing the electrons into sets of two groups , 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 (αJ1, βJ2) J, with J1 and J2 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 (αJ1, βJ2) subset are obtained by combining J1 and J2 in the usual way.