### 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) = ∑_{j}E′_{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 F_{m}(δ) = ∑_{j}E′_{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 language | English |
---|---|

Pages (from-to) | 5112-5122 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 39 |

Issue number | 10 |

Publication status | Published - Oct 1998 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*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.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 39, no. 10, pp. 5112-5122.

}

TY - JOUR

T1 - Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems

AU - Elout, Melchior O.

AU - Goodson, David Z.

AU - Elliston, Carl D.

AU - Huang, Shi Wei

AU - Sergeev, Alexei V.

AU - Watson, Deborah K.

PY - 1998/10

Y1 - 1998/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0032332524&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032332524&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032332524

VL - 39

SP - 5112

EP - 5122

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

ER -