Translation of STO charge distributions

Abstract

Barnett and Coulson's zeta-function method (M. P. Barnett and C. A. Coulson, Philos. Trans. R. Soc., Lond. A 1951, 243, 221) is one of the main sources of algorithms for the solution of multicenter integrals with Slater-type orbitals. This method is extended here from single functions to two-center charge distributions, which are expanded at a third center in terms of spherical harmonics times analytical radial factors. For s-s distributions, the radial factors are given by a series of factors corresponding to the translation of s-type orbitals. For distributions with higher quantum numbers, they are obtained from those of the s-s distributions by recurrence. After analyzing the convergence of the series, a computational algorithm is proposed and its practical efficiency is tested in three-center (AB/CC) repulsion integrals. In cases of large basis sets, the procedure yields about 12 correct significant figures with a computational cost of a few microseconds per integral.

Electric field integrals for Slater‐type orbitals

“…In previous work, we have examined different ways to obtain a general formula for V ( r ). After comparison of the performances obtained using ellipsoidal coordinates 16 and translation methods 17, we found the ellipsoidal coordinates to lead to a more robust and efficient algorithm, which we implemented in our STO integral package 5.…”

“…In this expression, ξ, η, and ϕ are the ellipsoidal coordinates of r in the lined‐up system, R ≡ | R I − R J |, β = (ζ i + ζ j ) R /2, ν = (ζ i − ζ j ) R /2. M + , M − , s + , s − , A and B are numerical constants, and: Stable and efficient algorithms for these integrals ( I , K , and L ) have been reported in references 16, 20.…”

“…In sections 3 and 5, general expressions are reported (valid for STO with arbitrary quantum numbers) and, in sections 4 and 6, the simplifications corresponding to the field at the nuclei are discussed. It is also proved that their calculation can be facilitated through the use of some algorithms previously developed for overlap 15 and nuclear attraction integrals 16, 17. Finally, section 7 includes several examples as well as a discussion on the efficiency of the global procedure.…”

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.…”

Accuracy of the electrostatic theorem for high‐quality Slater and Gaussian basis sets

“…In recent works, we have used the theorem in the interpretation of both binding 4 and weak chemical forces 5 and found that, with high‐quality basis sets, it is fulfilled with sufficient accuracy for these applications. This suggests that the situation may be not as disastrous as the previous studies 1–3 indicate.…”