  The $4f-5d$ transition rates for rare-earth ions in crystals can be
calculated with an effective transition operator acting between model $4f^N$
and $4f^{N-1}5d$ states calculated with effective Hamiltonian, such as
semi-empirical crystal Hamiltonian. The difference of the effective transition
operator from the original transition operator is the corrections due to mixing
in transition initial and final states of excited configurations from both the
center ion and the ligand ions. These corrections are calculated using
many-body perturbation theory. For free ions, there are important one-body and
two-body corrections. The one-body correction is proportional to the original
electric dipole operator with magnitude of approximately 40% of the uncorrected
electric dipole moment. Its effect is equivalent to scaling down the radial
integral $\ME {5d} r {4f}$, to about 60% of the uncorrected HF value. The
two-body correction has magnitude of approximately 25% relative to the
uncorrected electric dipole moment. For ions in crystals, there is an
additional one-body correction due to ligand polarization, whose magnitude is
shown to be about 10% of the uncorrected electric dipole moment.
