Clebsch-Gordan coefficient symmetry properties

Rearranging the electron creation operators in the right side of this expression and using the symmetry properties of Clebsch-Gordan coefficients... 

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

Further, one must pay attention to Eq. (5.10) for /mm - In isotropic collisions, when we have Tmm> — ram=m-M i and in the absence of an external field, when we have jkjJmm1 — 0) linearly polarized excitation yields /mm i = -f-M i-M- And since /mm i and f-M i-M enter into the sum (5.41) with equal coefficients, the aforesaid implies the absence of transversal orientation f v More precisely, the contention concerning the antisymmetry of the respective density matrix elements /mm i = —f-M i-M follows from the explicit form of the cyclic components of the polarization vector (see Appendix A), and from the symmetry properties of the Clebsch-Gordan coefficients (see Appendix C). 

Clebsch-Gordan coefficients (C.3), that the magnitudes Ll5L2L-3 possess certain symmetry properties ... 

Sometimes, instead of Clebsch-Gordan coefficients, one uses the Wigner 3j-symbols which possess simpler symmetry properties. These are con-... 

We form the tensor product (3.99) of the two operators, using (3.93) to express the Clebsch—Gordan coefficient as a 3-j symbol and using the symmetry properties of the 3-j symbol. 

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