1 Introduction

  Multipole expansions and expansions in spherical harmonics are prominent examples of orthogonal expansions and play an important rôle in the sciences. Consequently, there is an immense literature regarding this topic. In chemistry, such expansions are important

• for the transitions between molecular states induced by an interaction with electromagnetic radiation in the context of time-dependent perturbation theory [1], Chap. 16,
• for the description of NMR (Nuclear Magnetic Resonance) experiments [2],
• for the description of intermolecular interactions that are fundamental for the classical description of many-body systems, for instance in terms of induced multipole moments like polarizability [3], Chap. 15
• in particular for the computation of the electrostatic potential of molecular charge distributions, e.g., by solving the Poisson equation [4], with applications also in density functional programs (deMon),
• for the characterization of molecules by electrostatic multipole moments (dipole, quadrupole, octupole and hexadecapole moments) where the first non-vanishing multipole moment is independent of the choice of the origin [5] that is in most cases chosen to be the center of mass of the molecule
• for the determination of effective atomic charges (partial charges) [6], [7] for force-field, molecular mechanics and molecular modeling calculations by fits to the electrostatic potentials of molecules [8], [9], [10], [11], [12], [13], [14] as an alternative to charges derived from population analysis [15], [16], [17], [18] or also to empirical charges that are derived on the basis of interaction energies and distances (CHARMm, [19], [20]) or properties of fluids [21], [22], [23], [24], [25], [26], [27], [28],
• for determining effective atomic multipole moments, also for molecular force-field calculation, where the effective forces between two molecules are represented as a sum of electrostatic interactions of such distributed atomic multipole moments of pairs of atoms [29], [30]
• in the Fast Multipole Method (FMM) of Greengard and Rokhlin [31], [32], [33], [34], [35], [36], [37], [38], [39], where the computational effort -- like in some other methods, compare the next item -- scales linearly with the number of atoms and which is currently used for molecular dynamics calculation of macro molecules with a complete description of long-range Coulomb interactions [40], [41], [42], [43] and will be part of the new quantum chemistry program Q-Chem of Johnson, Gill and Head-Gordon ,
• in the Distributed Parallel Multipole Tree Algorithm (DPMTA) where the computational effort also scales linearly with system size and that also has been used in molecular dynamics simulations of macro molecules ,
• in tight-binding Hartree-Fock calculations of polymers [44],
• for the calculation of molecular integrals (mostly with exponential-type basis functions) [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], in particular in combination with addition theorems and/or one-center expansions. [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102],[103], [104], [105], [106], [107], [108], [109], [110], [111], [112], [113], [114], [115], [116], [117]

These expansions can be regarded as generalized Fourier series. Since such expansions often converge rather slowly, there is a need for methods to accelerate the convergence. The basic approach is to transform the sequence of partial sums of the series into a new sequence that converges faster, while using only very simple arithmetics. As we will see, such methods can also improve drastically the convergence even in cases where the original series is already converging relatively fast. Such methods may even be used to calculate a meaningful value of divergent series. The additional calculational effort for the calculation of the sequence transformation is very low, and may often be neglected in comparison to the calculation of the terms of the series.

Not many successful methods for the convergence acceleration of Fourier series and orthogonal expansions are known. Some new methods have been introduced recently by the author [118], [119], [120], [121], [122], [123], [124], [125]. Methods for the convergence acceleration of expansions in orthogonal polynomials will be discussed in Sec. 4.

We study here the convergence acceleration of a one-center multipole expansion in comparison to an expansion of the exact electrostatic potential in spherical harmonics.

An extension of the results to the convergence acceleration of distributed multipole expansions where the electrostatic potential is represented by a sum of truncated multipole expansions at various centers (mostly the atoms), can simply be obtained by separate convergence acceleration of each of these expansions and subsequent summation.

As a first step, we limit attention to expansions in Legendre polynomials that arise for rotationally symmetric problems. It is believed that more general types of expansions can be treated similarly. This, however, is still under investigation.

First, we will recall some basic facts on multipole expansions of electrostatic potentials. Then, we go on to point out the connection to three-center nuclear attraction integrals. Before going on to a specific example, we discuss methods for convergence acceleration of such orthogonal expansions based on nonlinear sequence transformations. As a simple example, we treat a two-center density that could arise in molecular LCAO calculations. Numerical tests are presented that show that the extrapolated values obtained by using the nonlinear sequence transformation converge much faster than the original expansions, both for the multipole expansion of the potential, and for its exact expansion in spherical harmonics.