Improvements on the application of convergence accelerators for the evaluation of some three-electron atomic integrals

Abstract

Convergence accelerator methods are employed to analyze some of the most difficult three-electron integrals that arise in atomic calculations. These integrals have an explicit dependence on the interelectronic coordinates, and take the form integral r(i)(1)r(j)(2)r(k)(3)r(l)(23)r(m)(31)r(n)(12) exp((-alpha(r1)-beta(r2)-gamma(r3))dr(1)dr(2)dr(3). The focus of the present investigation are the most difficult cases of the parameter set [i, j, k, l, m, n]. Several convergence accelerator techniques are studied, and a comparison presenting the relative effectiveness of each technique is reported. When the convergence accelerator approach is combined with specialized numerical quadrature methods, we find that the overall technique yields high-precision results and is fairly efficient in terms of computational resources. Difficulties associated with the standard numerical precision loss of convergence accelerator techniques are circumvented.

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

“…Also, we should mention that more numerical examples have already been given, e.g., in Refs. [34,148,152]. Here, we discuss a few other examples in depth.…”

Extended Hylleraas three-electron integral

“…For comparison with the former results obtained in Ref. [3], we have taken the same n i and w i as in Table II Considering the extended Hylleraas integral with r −2 1 , we calculate it by numerical integration with respect to w 1…”

An algorithm for calculating atomic D states with explicitly correlated Gaussian functions

“…Part of the reason for the lack of calculations of these states is due to the basis set, which for most works concerning atomic levels has involved Hylleraas type functions. These functions, while very effective in the calculations of two-and three-electron atoms, [2][3][4][5][6][7] have not yet been extended to four-electron atoms due to difficulties with calculating the Hamiltonian matrix elements.…”