Resolving a distribution of charge into intrinsic multipole moments: A rankwise distributed multipole analysis

We present a method for the rankwise distributed multipole analysis of an arbitrary distribution of charge and its surrounding field. Using the superposition principle, the electrostatic field created by a distribution of charge can be resolved recursively into the contributions of a set of intrinsic multipole moments “tied to” their rank-specific multipole centers. The positions of the multipole centers, which are fixed with respect to the distribution of charge, are determined from a term-by-term optimization of the Taylor’s expansion of the electrostatic potential with respect to the charge coordinates. We describe the recursive construction of the intrinsic multipole moments and derive the algebraic expression of the multipole centers. The resulting distributed multipole expansion provides a conceptual framework for the analysis and modeling of the electrostatic field and of its associated distribution of charge. I. INTRODUCTION Multipole expansions are a central conceptual and computational technique in the analysis of the electrostatic and, more generally, of any potential field in terms of the spatial distribution of its source. The importance of the method is due in part to the relationship that it establishes between the field “outside” the source and the macroscopic description of the source of field provided by its set of multipole moments. However, the method poses a number of limitations, from both a conceptual and a practical point of view. Here, we address some of these limitations. The coefficients of the multipole expansion form the components of the multipole moments of the spatial distribution of charge. Except for the lowest-rank nonvanishing multipole, all other moments depend on the center of expansion. Therefore, the expansion is not uniquely defined, depending instead on the expansion center. There is no general guiding principle for choosing a “best” expansion center even though, in practice, it is common to use the center of charge, or the center of mass when the total charge is zero and the center of charge is not defined. In this paper we propose a multipole expansion that removes this ambiguity. In principle, the center of expansion is not important in the sense that, when the whole multipole series is retained, the potential converges to the exact value, independent of the location of the center of expansion. Moreover, in many practical applications of the multipole expansions, the field is needed at large distances from the source of charge. In those cases, a good approximation can be obtained by truncating the multipole series to a relatively low number of terms, and the location of the center of expansion is not of much importance. In other applications, however, the assumption of a large distance from the observation point to the source is not warranted. In these latter cases, the truncation errors may be very sensitive to the location of the center of expansion, and an optimization of the center of expansion is desirable so that the rate of convergence is maximized within the retained terms. Then, the reasonable question arises: how does the error in this approximation of the potential depend on the origin of expansion and, related to this, how can one minimize this error? In part, addressing this question led to the development of more specific methods for the use of the multipolar expansions in various physical for example, the fast multipole method FMM1 and physical chemistry applications for example, the distributed multipole analysis DMA2,3. We also address this question with the analysis presented here. However, unlike previous methods that are specifically tailored for certain types of applications, the solution we propose is general, being based on intrinsic properties of the multipole moments alone. This second aspect is especially important for the appli