Alternative basis functions forL2calculations on the molecular continuum. II. Integrals with higher-order functions

Abstract

By differentiation of the expressions for the basic s-type integrals previously presented, analytic expressions are here derived for integrals, relevant for quantum-chemistry calculations, involving oscillating Hermite Gaussian functions (OHGF's) and many-center Hermite Gaussian functions (HGF's) of any order. The OHGF is the product of a HGF and a radial trigonometric factor cos(kr), and has been proposed for describing the continuum orbitals in L calculations on molecules. The resulting expressions are compact and particularly suitable for numerical implementation on a computer, while the increase in the computational effort of the integral evaluation with respect to the s-type functions is estimated to be of the same order as that found in the standard case of simple Gaussian functions. Applications of the OHGF basis to the calculation of continuum states are presented and compared to the exact results for test cases: the hydrogen atom and the H, + molecule, in order to discuss advantages and limitations of the proposed approach.PACS number(s): 31.15. +q, 02.60.+y

Alternative basis functions forL2calculations on the molecular continuum. I. The basic prototype integrals

“…These basis functions combine the oscillating character of a trigonometric factor [cos(kr)] with the "transferability" of the HGF's and therefore represent a convenient basis set for such calculations. The quality of the description of the continuum orbital can be considered comparable with that already obtained by the STOCOS's [9] (as explicitly shown in a following paper [14]), with the advantage of not being limited to a particular class of molecules. Such a description is therefore [7] much superior to that offered by spherically averaged plane waves [12] or pure Gaussian sets [13], which latter can represent only a limited number of radial nodes because of redundancy problems that can be easily avoided using a set of OHGF's.…”

“…However, numerical tests have shown that a basis set of OHGF's and many-center HGF's can be considered competitive with the ones currently employed even by the direct application of the expressions here presented. In a following paper [14] we will derive the formulas for the integ rais involving higher-order functions; however, we note that the s-type functions considered here can be directly utilized as such in molecular calculations by adopting a "lobe-Gaussianfunction" approach [16].…”

“…For the bound orbitals we suggest the use of many-center polynomial Gaussian functions as in the standard molecular calculations and therefore in Sec. III we derive analytic expressions for the prototype integrals, relevant in the quantum-chemistry methods, involving s-type OHGF's and many-center s-type HGF's (higher-order functions will be considered in a following paper [14], hereafter referred to as II, where applications to test problems will also be presented). Finally in Sec.…”

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

“…Therefore, we will now discuss those properties of \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{Y}^{m}_{\ell}(\nabla)$\end{document} that are needed for the derivation of addition theorems for B functions. Other properties as well as numerous applications—mainly in the context of multicenter integrals—are described in the literature 6, 19, 29, 55, 57, 93, 105–135.…”

Addition theorems as three-dimensional Taylor expansions

“…In this section, those properties of the spherical tensor gradient operator 𝒴(∇) are reviewed that are needed for the derivation of addition theorems. Other properties as well as numerous applications—mainly in the context of molecular multicenter integrals—can be found in the literature 43, 55, 65, 74, 84, 86–112.…”