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Perturbative contributions to single-beam two-photon transition rates may be divided into two types. The first, involving low-energy intermediate states, require a high-order perturbation treatment, or an exact diagonalization. The other, involving high energy intermediate states, only require a low-order perturbation treatment. We show how to partition the effective transition operator into two terms, corresponding to these two types, in such a way that a many-body perturbation expansion may be generated that obeys the linked cluster theorem and has a simple diagrammatic representation., Comment: 11 preprint pages

The $f\to d$ transition model by Duan and co-workers [Phys. Rev. B {\bf 66}, 155108 (2002); J. Solid State Chem. {\bf 171}, 299 (2003)] has been very useful in interpreting the $f\to d$ absorption, emission and nonradiative relaxation of light lanthanide ions in crystals. However, based on the assumption that the $f^{N-1}$ core spin-orbit interaction is weak compared to $f\to d$ exchange interaction, this model, in the original form, is not applicable to interpretation of the $f\to d$ transitions of heavy lanthanide ions or actinide ions in crystals. In this work the model is extended to cover the cases of heavy lanthanides and actinides, where the spin-orbit interaction of $f$ orbitals may be stronger than the $f-d$ exchange nteraction. Keywords: lanthanide; f-d transition; model; spectrum; actinide; VUV, Comment: Presented on AMN-2, Queenstown, New Zealand Feb 6-11, 2005; Will appear on Current Applied Physics

Effective operators needed in single-beam two-photon transition calculations have been represented with modified Goldstone diagrams similar to the type suggested by Duan and co-workers [J. Chem. Phys. 121, 5071 (2004) ]. The rules to evaluate these diagrams are different from those for effective Hamiltonian and one-photon transition operators. It is verified that the perturbation terms considered contain only connected diagrams and the evaluation rules are simplified and given explicitly., Comment: 10 preprint pages, to appear in Journal of Alloys and Compounds