IMPROVING THE CONVERGENCE AND ESTIMATING THE ACCURACY OF SUMMATION APPROXIMANTS OF 1 D EXPANSIONS FOR COULOMBIC SYSTEMS/

The convergence of large-order expansions in delta = 1/D, where D is t he dimensionality of coordinate space, for energies E(delta) of Coulom b systems is strongly affected by singularities at delta = 1 and Pade' -Borel approximants with modifications that that completely remove the singularities at delta = 1 and remove the dominant singularity at del ta = 0 are demonstrated. A renormalization of the interelectron repuls ion is found to move the dominant singularity of the Borel function F( delta) = Sigma(j)E(j)'/ j!, where E-j' are the the expansion coefficie nts of the energy with singularity structure removed at d51, farther f rom the origin and thereby accelerate summation convergence. The groun d-state energies of He and H-2(+) are used as test cases. The new meth ods give significant improvement over previous summation methods. Shif ted Borel summation using F-m(delta) = Sigma(j)E(j)'/Gamma(j + 1 - m) is considered. The standard deviation of results calculated with diffe rent values of the shift parameter m is proposed as a measure of summa tion accuracy. (C) 1998 American Institute of Physics. [S0022-2488(98) 04210-8].