Unexpected relations between matrix elements of the three-body scalar operators ti in the f shell

It has been known for some time that crystal-field matrix elements (i.e., matrix elements of sums over spherical harmonics involving the coordinates of the individual electrons) are often unexpectedly proportional to one another in the f shell. To see whether similar relations hold for more complicated operators than those provided by the crystal field, we examined the matrix elements of the three-electron scalar operators t_i for all configurations f^N, as calculated by W. T. Carnall on the basis of the computer program of Hannah Crosswhite. These operators are widely used to take configuration interaction into account, and we found a surprising number of proportionalities that go beyond what would be expected on a straightforward application of the Wigner-Eckart theorem, as applied to the irreducible representations of the classic groups SO(7), G₂ and SO(3) used by Racah in defining the f-electron states. A listing of such relations is provided.