L Journal of Alloys and Compounds 344 (2002) 272–275 www.elsevier.com / locate / jallcom Effective two-photon transition operators: perturbative calculations and connectivity of diagrams * C.K. Duan , M.F. Reid University of Canterbury, Department of Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand Abstract Perturbation theory expansions appropriate to two-photon processes between energy levels of rare earth ions are examined. The construction of effective two-photon transition operators that contain only connected diagrams is shown to be possible, but more complicated than previously thought.  2002 Elsevier Science B.V. All rights reserved. Keywords: Two-photon process; Rare earth ions; Transition operators; Perturbation theory 1. Introduction sented by Goldstone diagrams [15,16]. The linked cluster theorem [14,15,17–19] implies that disconnected diagrams N The theory of two-photon absorption within the 4f cancel for effective Hamiltonians and effective operators, configuration of rare earth ions was developed by Axe [1] provided that the model space has been carefully chosen in 1964. This theory was a zeroth order theory in the sense and the model states have been properly orthonormalised. that no perturbation of the wave functions was considered. This cancellation reduces the number of high order dia- Much later perturbations involving the spin–orbit [2], grams greatly and ensures size consistency. crystal field [3,4] and the Coulomb interactions [5,6] were The application of MBPT to two-photon transitions is N considered. Two-photon absorption between the 4f and more difficult than the case of one-photon transitions N21 4f 5d configuration has also been measured [7–9] and because of the energy denominators that contain both the calculated [10,11]. These transitions involve odd-parity photon energies and exact electronic energies. It seems crystal-field mixing of the configurations, in analogy to reasonable to expect that a connected expansion for the N one-photon electric-dipole transitions within the 4f con- two-photon effective transition operator should exist, but figuration. this has not been explicitly demonstrated. It is obvious that The dominant contributions to two-photon transitions a connected expansion for an effective operator is only usually arise from the electric dipole moments. However, it possible if the operator itself is connected. In this paper we is possible to have important magnetic dipole moment show that this connectedness is not trivial, but can be contributions. Evidence for the contribution of magnetic- demonstrated order by order. dipole moments is provided by the two-photon circular dichroism measurements and calculations of Richardson and co-workers [12,13]. 2. Zeroth order approximation of the two-photon Many-body perturbation theory (MBPT) has been ap- transition operator and connectivity plied to calculations of energy levels and transition inten- sities in a wide variety of materials [14,15]. The key Two-photon transition rate from an initial state ui l to a concept in MBPT is that effective Hamiltonians and N final state u f l via all possible virtual intermediate states ukl effective operators act within a model space (such as 4f ) is proportional to the product of the intensities of the two to give exact energy levels and matrix elements. These optical beams and the square of the following quantity: operators are usually expanded order by order and repre- k f uD uklkkuD ui l k f uD uklkkuD ui l 2 1 1 2 *Corresponding author. Fax: 164-3-364-2469. ]]]]] ]]]]] O 1 (1) S D E 1 v 2 E E 1 v 2 E i 1 k i 2 k E-mail address: c.duan@phys.canterbury.ac.nz (C.K. Duan). k 0925-8388 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388(02)00367-5