Pade summation approximation

Starting from eqn (28) and applying the Pade summation approximation, it is possible to define a family of correlation functionals, which has the general form ... 

Figure 13. Accuracy of Borel summation for the ground state energy of Hj with R = 1. The cmwes axe labeled as follows solid curve, Pade-Borel approximants dashed curve, Darboux-Borel ap-proximants. Hybrid Fade summation is shown, for comparison, as the dotted curve. All three curves were calculated using exact values for both residues. The number of accurate digits is defined as — logio KS n — E)/E, where E is the exact energy.

The computed complex energy eigenvalues E = F, — T/2 for different states ni, ri2, m) of a hydrogen atom are presented in Table 2, where (1/n) denotes the results of the 1/n-expansion and HPA marks the values obtained by perturbation series summation with the help of Hermite-Pade approximations [6]. The agreement between different computational methods is quite good. 

We can improve the Pade-Borel summation by explicitly incorporating a square root in the approximant for the integrand [9,10]. We determined in the last Section that the large-order behavior of F 8) for Hj was determined by a singularity of the form a(l-f This... 

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