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

TL;DR

This paper develops a many-body perturbation theory method to accurately calculate $4f-5d$ transition rates in rare-earth ions, accounting for state mixing and ligand effects, improving theoretical predictions for these optical transitions.

Contribution

It introduces a comprehensive calculation framework using many-body perturbation theory to include corrections from state mixing and ligand polarization in transition rate calculations.

Findings

One-body correction scales the electric dipole moment by about 40%.

Two-body correction contributes approximately 25% of the uncorrected dipole moment.

Ligand polarization adds an extra 10% correction for ions in crystals.

Abstract

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.