Journal of Computational Physics 180, 642–658 (2002) doi:10.1006/jcph.2002.7110 A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions Leslie F. Greengard ∗,1 and Jingfang Huang† ,2 ∗ Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012; and †Department of Mathematics, University of North Carolina at Chapel Hill, CB 3250 Phillips Hall, Chapel Hill, North Carolina 27599 Received May 22, 2001; revised May 1, 2002 We present a new version of the fast multipole method (FMM) for screened Coulomb interactions in three dimensions. Existing schemes can compute such in- teractions in O( N ) time, where N denotes the number of particles. The constant implicit in the O( N ) notation, however, is dominated by the expense of translating far-field spherical harmonic expansions to local ones. For each box in the FMM data structure, this requires 189 p 4 operations per box, where p is the order of the expansions used. The new formulation relies on an expansion in evanescent plane waves, with which the amount of work can be reduced to 40 p 2 + 6 p 3 operations per box. c  2002 Elsevier Science (USA) Key Words: translation operators; fast multipole method; screened Coulomb interaction. 1. INTRODUCTION In the last few years, new versions of the fast multipole method (FMM) have been developed for the evaluation of harmonic potential fields in three dimensions. The schemes of [4, 10], for example, are extremely efficient in the evaluation of pairwise interactions in large ensembles of particles: (x j ) = N  i =1 i = j q i ‖x j - x i ‖ , (1) 1 This work was supported in part by the Applied Mathematical Sciences Program of the U.S. Department of Energy DEFGO288ER25053. 2 Fax: (919) 962-9345. E-mail: huang@amath.unc.edu. The work of this author was supported by the Applied Mathematical Sciences Program of the U.S. Department of Energy under Contract DE-FGO288ER25053. 642 0021-9991/02 $35.00 c  2002 Elsevier Science (USA) All rights reserved.