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The $4f-5d$ transition rates for rare-earth ions in crystals can becalculated with an effective transition operator acting between model $4f^N$and $4f^{N-1}5d$ states calculated with effective Hamiltonian, such assemi-empirical crystal Hamiltonian. The difference of the effective transitionoperator from the original transition operator is the corrections due to mixingin transition initial and final states of excited configurations from both thecenter ion and the ligand ions. These corrections are calculated usingmany-body perturbation theory. For free ions, there are important one-body andtwo-body corrections. The one-body correction is proportional to the originalelectric dipole operator with magnitude of approximately 40% of the uncorrectedelectric dipole moment. Its effect is equivalent to scaling down the radialintegral $\ME {5d} r {4f}$, to about 60% of the uncorrected HF value. Thetwo-body correction has magnitude of approximately 25% relative to theuncorrected electric dipole moment. For ions in crystals, there is anadditional one-body correction due to ligand polarization, whose magnitude isshown to be about 10% of the uncorrected electric dipole moment.Comment: 10 pages, 1 figur

General calculation of 4f-5d transition rates for rare-earth ions using many-body perturbation theory