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Journal of Mathematical Physics -- October 1998 -- Volume 39, Issue 10, pp. 5112-5122

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Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems

Melchior O. Elout, David Z. Goodson, Carl D. Elliston, Shi-Wei Huang, and Alexei V. Sergeev

Department of Chemistry, Southern Methodist University, Dallas, Texas 75275

Deborah K. Watson

Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019

(Received 30 March 1998; accepted 4 May 1998)

The convergence of large-order expansions in = 1/D, where D is the dimensionality of coordinate space, for energies E() of Coulomb systems is strongly affected by singularities at = 1 and = 0. Padé–Borel approximants with modifications that completely remove the singularities at = 1 and remove the dominant singularity at = 0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F() = _jE/j!, where E are the the expansion coefficients of the energy with singularity structure removed at = 1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using F_m() = _jE/(j + 1 – m) is considered. The standard deviation of results calculated with different values of the shift parameter m is proposed as a measure of summation accuracy. © 1998 American Institute of Physics.

PII: S0022-2488(98)04210-8
PACS: 31.10.+z, 02.30.Lt      Additional Information

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