Abstract. The interaction between multipoles is not isotropic even in cubic systems. This results in the introduction of geometric reduction factors in the calculation of energy-transfer rates in crystals. We derive these reduction factors for the cases of dipole± dipole, dipole±quadrupole, and quadrupole±quadrupole couplings and present a general procedure for their derivation in other cases. For the dipole±dipole case the geometric factor is independent of the distribution of the acceptor species, but for higher-order couplings, a signi®cant angular dependence is found. Key words: Energy transfer ± Shell model ± Multipolar interaction ± Angular dependence ± Elpasolite crystal 1 Introduction In a recent series of papers we derived a discrete shell model for energy transfer and applied this model to the luminescence decay curves of excited electronic states of rare-earth ions in the cubic hexachloroelpasolite crystal Cs 2 NaLnCl 6 [1±3]. In the systems considered in these papers the dominant mechanism of energy transfer is dipole±dipole coupling. Direct calculation using a Car- tesian basis shows that, in this case, the energy-transfer rate is independent of the precise angular distribution of the acceptors around the donor although, of course, it does depend on the number and distance of these acceptors. In some cases we found experimentally that the interaction between donors and acceptors was apparently of shorter range than the R 6 dependence of a dipole±dipole interaction and this led us to consider the possibility of both higher-order couplings and the eect of a nonisotropic dielectric shielding factor. To consider these possibilities quantitatively it is necessary to calculate the eect of the angular dependence of the coupling of higher multipoles on the energy-transfer rates. This paper describes a method for calculating these quantities. 2 The shell model for energy transfer In the presence of a single chemical type of optically active acceptor for the donor excitation energy, the energy-transfer processes are due to cross relaxation and within the shell model the luminescence decay curves following a d-function excitation pulse take the form [1] I t I 0 expk 0 t Y shells n1 X N n r n 0 O N n r n x  exp G s p n R 1 R n   s p r n k CR t   : 1 x is the mole fraction of the optically active ion, R n is the distance between the donor ion and an acceptor ion in the nth shell determined by the crystal structure, k 0 is the intrinsic decay rate involving radiative and nonradiative single-ion processes, and k CR is the cross-relaxation rate from a donor ion to a single acceptor in the ®rst shell. s p = 6, 8, or 10 for electric dipole vibronic±electric dipole vibronic (EDV±EDV) or magnetic dipole±mag- netic dipole (MD±MD), electric dipole vibronic±electric quadrupole (EDV±EQ), and electric quadrupole±electric quadrupole (EQ±EQ) interactions, respectively. The index p distinguishes between dierent coupling compo- nents for a given value of s. In octahedral symmetry the electric and magnetic dipole operators transform as the irreducible representation (irrep) T 1 and therefore only a single component exists for dipole±dipole coupling. The EQ operator transforms as the direct sum E  T 2 and three (six) dierent components occur for EDV±EQ (EQ±EQ) coupling. In our shell model we assume FoÈrster±Dexter multi- pole±multipole interaction among donor and acceptor ions with the resonant cross-relaxation rate given by [4, 5] Correspondence to: T. Luxbacher Regular article The angular dependence of the multipole±multipole interaction for energy transfer Thomas Luxbacher 1 , Harald P. Fritzer 1 , James P. Riehl 2 , Colin D. Flint 3 1 Institut fuÈr Physikalische und Theoretische Chemie, Technische UniversitaÈt Graz, Rechbauerstrasse 12, A-8010 Graz, Austria 2 Department of Chemistry, Michigan Technical University, 1400 Townsend Drive, Houghton, MI 49931, USA 3 Laser Laboratory, Department of Chemistry, Birkbeck College, University of London, 29 Gordon Square, London WC1H 0PP, UK Received: 6 November 1998 / Accepted: 15 January 1999 / Published online: 7 June 1999 Theor Chem Acc (1999) 103:105±108 DOI 10.1007/s002149900003