Calculation of single-beam two-photon absorption transition rate of   rare-earth ions using effective operator and diagrammatic representation
  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. 
Chang-Kui Duan, Michael F. Reid, Gang Ruan and hui-Ning Dong