Spin-orbit matrix elements in the atomic f shell from automorphisms of SO(8)

A scrutiny of the matrix elements within the f shell of the spin-other-orbit interaction H(soo) has revealed many unexpected proportionalities that go beyond an application of the Wigner-Eckart theorem to Racah's groups G2 and SO(7). An explanation is sought by using the automorphisms of SO(8) by means of which the states of the atomic f shell are generated by two alternative SO(7) bases of the type (1 / 21 / 21 / 2)(4), each augmented by two parity labels. Transformations between the bases can be made by reversing the relative phases of the two angular-momentum states 3 and 0 that comprise any one of the four SO(7) spinors. The method is exemplified by the single-electron spin-orbit interaction H(so), for which a component can be found that is invariant under the phase reversal. The extension from H(so) to H(soo) is described for the component z(6) of H(soo), and several examples drawn from the comparatively inaccessible configurations near the middle of the f shell are presented.