Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

Using expansion formula for the translation of Slater-type orbitals (STOs) obtained with the help of Coulomb Sturmian exponential-type functions (ETFs) (Guseinov, I. I. Int J Quantum Chem 2001, 81, 126) the multicenter electron-repulsion integrals are expressed in terms of the basic overlap integrals with the same screening parameters and the two-center basic hybrid integrals. For the calculation of basic hybrid integrals the recurrence relations are derived. These recurrence relations enable us to calculate some basic hybrid integrals by the use of which other basic integrals are calculated analytically from the formulas in terms of new auxiliary functions introduced recently by the authors. The recurrence relations obtained in this work for basic hybrid integrals are especially useful for the calculation of multicenter electron-repulsion integrals for large quantum numbers of STOs.

Slater Type Orbitals (STOs) which one of the types of exponential type orbitals (ETOs) are used usually as basis functions in the multicenter molecular integrals to better understand physical and chemical properties of matter. In this work, we develop algorithms for two-center overlap and two-center two-electron hybrid and Coulomb integrals which are calculated with help of translation method for STOs and some auxiliary functions by V. Magnasco's group. We use Mathematica programming language to produce algorithms for these calculations. Numerical results for some quantum numbers are presented in the tables. Consequently, we compare our obtained numerical results with the other known literature results and other details of evaluation method are discussed.

By the use of exponential-type functions (ETFs) the simpler formulas for the expansion of Slater-type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001

Analytical properties of new complete orthonormal sets of Psi(alpha) exponential-type orbitals (Psi(alpha)-ETOs where alpha=1, 0, -1, -2, em leader ), introduced by the author as finite linear combinations of Slater-type orbitals (STOs), are studied. Addition and expansion theorems for Psi(alpha)-ETOs are obtained in both coordinate and momentum representations. Using expressions of Psi(alpha)-ETOs in terms of STOs, the new methods are suggested to calculate multicenter multielectron integrals over STOs.

A unified treatment of two-center overlap integrals over Slater-type orbitals (STO) with integer and noninteger values of the principal quantum numbers is described. Using translation and rotation formulas for spherical harmonics, the overlap integrals with integer and noninteger n Slater-type orbitals are expressed through the basic overlap integrals and spherical harmonics. The basic overlap integrals are calculated using auxiliary functions Aσ and Bk. The analytical relations obtained in this work are especially useful for the calculation of overlap integrals for large integer and noninteger principal quantum numbers. The formulas established in this study for overlap integrals can be used for the construction of series expansions based on addition theorems. PACS Nos.: 31.15.–p, 31.20.Ej

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

Symbolic calculation is applied to the evaluation of molecular integrals over Slater orbitals (STOs). A recurrence scheme is developed for some new auxiliary functions that facilitate this work. Closed expressions are developed independently for use in spot checks. A table of formulas for the individual functions has been constructed and made available on-line. Applications are discussed. The derivations and tabulations were performed mechanically in runs that produced both the original typeset account of this study and more detailed documentation without manual transcription. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 464–472, 2000

Chapter 9 - Reducing and Solving Electric Multipole Moment Integrals

Calculation of two-center nuclear attraction integrals of Slater type orbitals with noninteger principal quantum numbers using Guseinov’s one-center expansion formulas and Löwdin-α radial function

Hartree‐Fock‐Roothaan (HFR) calculations for ground states of some atoms, i.e. He, Be, Ne, Ar, and Kr have been performed using minimal basis sets of Slater type orbitals (STOs) with integer and noninteger principal quantum numbers (integer n‐STOs and noninteger n‐STOs). The obtained total energies for these atoms using minimal basis sets of integer n‐STOs are in good agreement with those in the previous literature. On the other hand, for the case of minimal basis sets of noninteger n‐STOs, although the calculated total energies of these atoms agree well with the results in Literature, some striking results have been obtained for atoms Ar and Kr. Our computational results for the energies of atoms Ar and Kr are slightly better than those in literature. by amount of 0.00222 and 0.000054 a.u., respectively. The improvement in the energies of atoms Ar and Kr may result from the efficient calculations of one‐center two‐electron integrals over noninteger n‐STOs. For some atomic ions in their ground state, HFR calculations have been carried out using minimal basis sets of noninteger n‐STOs. The obtained total energies for these atomic ions are substantially lower than those available in literature.

In this paper, derivation of analytical expressionsfor overlap integrals with the same and different screening parametersof Slater type orbitals (STOs) via the Fourier-transform methodis presented. Consequently, it is relatively easy to expressthe Fourier integral representations of the overlap integralswith same and different screening parameters mentioned as finitesums of Gegenbauer, Gaunt, binomial coefficients, and STOs.

Using expansion formulae for multicenter charge densities of integer and noninteger n Slater type orbitals (ISTO and NISTO) established by one of the authors with the help of unsymmetrical one-range addition theorems of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETO), where α is the frictional quantum numbers (α = 1,0, - 1, - 2, … ), we perform the calculations for one-electron multicenter integrals of Slater type orbitals (STO) and Coulomb–Yukawa like correlated interaction potentials (CIP) with integer and noninteger indices (ICIP and NICIP). The convergence of the series is tested by calculating concrete cases for arbitrary values of indices and screening constants of orbitals and potentials. It should be noted that the results for one-electron multicenter integrals presented in this work were not shown in our previous publications. As an example of application, the calculations are performed for the ground state of electronic configuration of molecule BH2 that has the open shell.

A unified treatment of one-electron two-center integrals over noninteger n Slater-type orbitals is described. Using an appropriate prolate spheroidal coordinate system with the two atomic centers as foci, all the molecular integrals are expressed by a single analytical formula which can be readily and compactly programmed. The analysis of the numerical performance of the computational algorithm is also presented.

The dipole moment integrals for Slater-type atomic orbitals are conveniently and concisely expressed in terms of the C functions introduced by Ruedenberg, Roothaan, and Jaunzemis in their treatment of the two-center hybrid integrals. Explicit formulas in terms of both the C functions and the usual A and B functions are given for all such integrals involving orbitals with quantum numbers one and two.

Non-inverse-square force–distance law for long thin magnets—Revisited