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, Alexei V. Sergeev, Deborah K. Watson

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

The convergence of large-order expansions in δ= 1/D, where D is the dimensionality of coordinate space, for energies E(δ) of Coulomb systems is strongly affected by singularities at δ= 1 and δ= 0. Padé-Borel approximants with modifications that completely remove the singularities at δ= 1 and remove the dominant singularity at δ= 0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F(δ) = ∑jE′j/j!, where E′j are the the expansion coefficients of the energy with singularity structure removed at δ= 1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H+ 2 are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using Fm(δ) = ∑jE′j/Γ(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.

Original languageEnglish
Pages (from-to)5112-5122
Number of pages11
JournalJournal of Mathematical Physics
Volume39
Issue number10
Publication statusPublished - Oct 1998
Externally publishedYes

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Summation
estimating
Singularity
expansion
Coulomb Systems
Borel Functions
Ground State Energy
Energy
Renormalization
Standard deviation
Accelerate
Dimensionality
energy
standard deviation
ground state
shift
Coefficient
coefficients

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Elout, M. O., Goodson, D. Z., Elliston, C. D., Huang, S. W., Sergeev, A. V., & Watson, D. K. (1998). Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems. Journal of Mathematical Physics, 39(10), 5112-5122.

Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems. / Elout, Melchior O.; Goodson, David Z.; Elliston, Carl D.; Huang, Shi Wei; Sergeev, Alexei V.; Watson, Deborah K.

In: Journal of Mathematical Physics, Vol. 39, No. 10, 10.1998, p. 5112-5122.

Research output: Contribution to journalArticle

Elout, MO, Goodson, DZ, Elliston, CD, Huang, SW, Sergeev, AV & Watson, DK 1998, 'Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems', Journal of Mathematical Physics, vol. 39, no. 10, pp. 5112-5122.
Elout, Melchior O. ; Goodson, David Z. ; Elliston, Carl D. ; Huang, Shi Wei ; Sergeev, Alexei V. ; Watson, Deborah K. / Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems. In: Journal of Mathematical Physics. 1998 ; Vol. 39, No. 10. pp. 5112-5122.
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