Effective Hamiltonians and Clebsch-Gordan coefficients in crystals

The matrix elements of the Luttinger-Bir-Picus $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ effective Hamiltonian are shown to be easily constructed as products of appropriate Clebsch-Gordan coefficients times symmetrized tensorial field quantities. This simplification should facilitate construction of Hamiltonians in case of a high degree of degeneracy by eliminating the need to calculate a large number of auxiliary angular momentum matrices. Illustrations in diamond $O_h^7$ and $\beta -\mathrm{W}$ structure $O_h^3$ are given.