Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators

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Abstract
The analytical representations for overlap, two-center nuclear attraction, and Coulomb integrals of B functions [E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)] can be divided into two categories: representations containing only a finite number of terms which, however, may contain canceling singularities for small internuclear distances and nearly equal scaling parameters, and infinite-series representations which do not contain canceling singularities but where enormous convergence problems may occur. In this paper we analyze the numerical properties of the infinite-series representations for the two-center integrals mentioned above. In our approach, we first analyze the convergence type of an infinite series by deriving asymptotic approximations for the terms of the series with large summation indices. The information which is gained by this asymptotic analysis can be used to choose the convergence accelerator which has optimal properties for the series under consideration. In this paper we only use two nonlinear convergence accelerators, the Shanks transformation, and Levin’s u transformation. The Shanks transformation, which essentially produces Padé approximants, is our choice in the case of linearly convergent monotonic series, whereas Levin’s u transformation leads to spectacular acceleration effects in the case of alternating series and in the case of logarithmically convergent series. According to our experience the use of nonlinear convergence accelerators greatly enhances the applicability of these infinite-series representations.

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