Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. II. Neumann expansion of the exchange integrals

Abstract

In this paper we consider the calculation of two-center exchange integrals over Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb interaction potential and consider calculation of all basic quantities which appear in the resulting expression. Analytical closed-form equations for all auxiliary quantities have already been known but they suffer from large digital erosion when some of the parameters are large or small. We derive two differential equations which are obeyed by the most difficult basic integrals. Taking them as a starting point, useful series expansions for small parameter values or asymptotic expansions for large parameter values are systematically derived. The resulting expansions replace the corresponding analytical expressions when the latter introduce significant cancellations. Additionally, we reconsider numerical integration of some necessary quantities and present a new way to calculate the integrand with a controlled precision. All proposed methods are combined to lead to a general, stable algorithm. We perform extensive numerical tests of the introduced expressions to verify their validity and usefulness. Advances reported here provide methodology to compute two-electron exchange integrals over STOs for a broad range of the nonlinear parameters and large angular momenta.

Transition moments between excited electronic states from the Hermitian formulation of the coupled cluster quadratic response function

“…The S X LM values of Chang and Tang were derived from A X LM with the experimental excitation energies. a Gaussian basis set: d-aug-cc-pVQZ 30,31 b Slater basis set: mg-dawtcc5d basis of Lesiuk et al 29,32,33 The XCC3(S) results are in a much better agreement with the results calculated with other theoretical methods than the results obtained with the XCC3(G) and with QRCC3(G) methods. The most dramatic improvement is observed for the 3 d 1 D−3 p 1 P • and 4 p 1 P • −3 d 1 D transitions.…”

“…Level a Gaussian basis set: d-aug-cc-pVQZ 30,31 b Slater basis set: mg-dawtcc4d basis of Lesiuk et al 29,32,33 with a similar number of basis function as the Gaussian basis set. c Slater basis set: mg-dawtcc5d basis of Lesiuk et al 29,32,33 the two methods. II that the CC3 approximation has a little effect on the transition strength values.…”

“…It is worth noting that the use of the Slater orbitals leads in some cases to substantially different results. a Gaussian basis set: d-aug-cc-pVQZ 30,31 b Slater basis set: mg-dawtcc5d basis of Lesiuk et al 29,32,33 Non-physical value. For details see section III D.…”

Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

“…The J s,q functions involve Appell's hypergeometric functions [44], (23) and their explicit forms are given as,…”

Efficient calculation of integrals in mixed ramp-Gaussian basis sets

“…The usefulness of non-Gaussian basis sets with improved cusp properties is illustrated most starkly by considering the current use 18 of Slater basis sets [19][20][21] for specific purposes despite the very long integral evaluation times, 22,23 as well as more generally in the Amsterdam Density Functional (ADF) program. 24 Thus, despite more than 80 yr of investigation, [25][26][27][28] research is still undertaken [29][30][31][32][33][34][35][36][37][38][39][40][41] to improve integral evaluation for Slatertype orbitals to make these calculations competitive with all-Gaussian calculations. Given this, mixed ramp-Gaussian basis sets arguably encapsulate the best of both worlds: characteristics similar to all-Slater basis sets with the potential to match or better all-Gaussian calculation speeds.…”