Auxiliary functions for molecular integrals with Slater‐type orbitals. I. Translation methods

Abstract

Many types of molecular integrals involving Slater functions can be expressed, with the -function method in terms of sets of one-dimensional auxiliary integrals whose integrands contain two-range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the -function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two-center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient.

Auxiliary functions for molecular integrals with Slater‐type orbitals. II. Gauss transform methods

“…It can be seen there that, although they are more expensive than the two-center integrals of the same type, which require less than 1 µs per integral [8], their cost is still very reasonable for practical applications. As a reference, we note that the integrals with l = 0 and l = 1 of this type computed with efficient codes based on ellipsoidal coordinates [25] require computational times about 3 µs but, in turn, the current code is not restricted to only these two values of l. …”

“…Moreover, integrals involving a two-center distribution of STO and an irregular harmonic placed at one of their centers were reported in a previous paper [8]. To complete the set of integrals of this type, we have written a FORTRAN code for the threecenter ones, following the lines of sections 4, 5, and 6.…”

“…In a recent work [8], we have analyzed several sets of auxiliary functions resulting from the application of the ζ -function method, and proved the formal and computational advantages of this study. In this work, we carry out a study of the functions that occur when the Gaussian transform is employed for solving several types of integrals.…”

“…Two-center integrals of this type (i.e. those with the irregular harmonic placed at one of the two centers) have been recently studied in the framework of the translation method [8]. We prove here that the three-center integrals (with the irregular harmonic placed at a third center) can be expressed in terms of auxiliary functions similar to those of the overlap and related integrals.…”

Combining Slater-type orbitals and effective core potentials

“…Interestingly enough, there has been a significant progress on this issue in recent years. In fact, looking at only the past 15 years, there are many notable works of Bouferguene et al [10][11][12][13], Rico et al [14][15][16][17][18][19][20][21][22][23][24], Hoggan et al [25][26][27][28][29][30], Pachucki [31][32][33][34][35], and others [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. In particular, for the diatomic systems STOs can be now used routinely [51].…”