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3-j symbol 4f configuration angle angular Angular-momentum Operators anticommutation apply axis basis functions C.L. Siegel character table Clebsch-Gordan coefficients commutation components configuration interaction conjugate corresponding creation operators CRYSTAL FIELD THEORY crystal quantum numbers crystalline potential cube degeneracy degenerate determinantal product double group Eckart theorem eigenfunction electric dipole electric-dipole energy levels equation example f electrons form a basis form basis free atom given Hamiltonian of Eq integral invariant irreducible representation Kramers Kronecker product label lattice Lemma linear combinations magnetic dipole transitions magnetic field magnetic-dipole momentum operators obtained occurs octahedral group octahedral symmetry odd parity operator equivalents parameters perpendicular phase factor plane radiation reduced matrix elements reflection representation H reversal rotation second quantization selection rules sides of Eq spherical harmonics spin functions spin-orbit coupling spin-orbit interaction split symmetry wave functions wavefunction Wigner Wigner-Eckart theorem write written