Harris J. Silverstone Convergence of the bipolar expansion for the coulomb potential International Journal of Quantum Chemistry 114
The bipolar expansion of the Coulomb potential, which underlies the multipole moment expansion for interacting charge distributions, converges like a geometric series for separated charges, but converges only conditionally when the charges interpenetrate. This article shows how the order of summation affects the sum. Evidence is also presented for the possibility of simpler limit series for the bipolar expansion when the geometry is linear.
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