Two-center Nuclear Attraction Integrals Research Articles

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

In this work, by the use of Guseinov’s one-center expansion formulas and Löwdin-α radial function, the series expansion relations in molecular coordinate system are established for the two-center nuclear attraction integrals of noninteger n∗ Slater type orbitals in terms of basic two-center nuclear attraction integrals over integer n Slater functions. The Löwdin α-radial function convoluted with the Guseinov’s one-center expansion formulas is one of the most important ingredients for accurate and efficient implementation of electronic structure calculation methods regardless of Hartree–Fock–Roothaan (HFR) method. The proposed algorithm shows better performance in arbitrary quantum numbers, screening constants and location of orbitals leading to significantly reduced run times.

Comment on “Analytical evaluation for two-center nuclear attraction integrals over Slater type orbitals by using Fourier transform method” by S. Özcan and E. Öztekin (J. Math. Chem. DOI 10.1007/s10910-008-9398-z)

S. Ozcan and E. Oztekin, (J. Math. Chem. doi:10.1007/s10910-008-9398-z) published formulas for evaluating the two-center nuclear attraction integrals over Slater type orbitals. It is shown that the analytical relations for these integrals through the expansion coefficients of the electron charge density for the one-center case and the overlap integrals presented in Sect. 3 of this work can easily be derived by means of a simple algebra from the formulas published in our papers (I.I. Guseinov, J Mol Struct (Theochem) 417:117, 1997; J Math Chem 42:415, 2007 and B.A. Mamedov, Chin J Chem 22:545, 2004). It should be noted that the formulas of overlap integrals presented by E. Oztekin et al., in previous paper (E. Oztekin, M. Yavuz, Ş. Atalay, J Mol Struct (Theochem) 544:69, 2001) for the calculation of two-center nuclear attraction integrals also are obtained from our papers (see Comment: I.I. Guseinov, J Mol Struct (Theochem) 638:235, 2003).

Analytical evaluation for two-center nuclear attraction integrals over slater type orbitals by using Fourier transform method

In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s.

Reply to “Comment on ‘Calculation of Two-center Nuclear Attraction Integrals over Integer and Noninteger N-slater-type Orbitals in Nonlined-up Coordinate Systems” ’

The comments of Guseinov on our recent paper (T. Özdoğan, S. Gümüş and M. Kara, J. Math. Chem. 33 (2003) 181) are critically analyzed. It is proved that the expansion formula for the product of two normalized associated Legendre functions and the expressions for two-center nuclear attraction integrals over Slater type orbitals have been obtained independently, by the use of basic mathematical rules.

Comment on “Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater-Type Orbitals in Nonlined-Up Coordinate Systems”

Recently published formulas for the calculation of two-center nuclear attraction integrals (T. Ozdogan, S. Gumus and M. Kara, J. Math. Chem. 33 (2003) 181) are critically analyzed. It is shown that the analytical relations presented in this work are not original and they can easily be derived by means of a simple algebra from the formulas for overlap integrals, their rotation coefficients and expansion of the product of two normalized associated Legendre functions in elliptical coordinates published in our papers (I.I. Guseinov, J. Phys. B 3 (1970) 1399; Phys. Rev. A 32 (1985) 1864; J. Mol. Struct.: Theochem 336 (1995) 17).

Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater Type Orbitals in Nonlined-Up Coordinate Systems

Two-center nuclear attraction integrals over Slater type orbitals with integer and noninteger principal quantum numbers in nonlined up coordinate systems have been calculated by means of formulas in our previous work (T. Ozdogan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The computer results for integer case are in best agreement with the prior literature. On the other hand, the results for noninteger case are not compared with the literature due to the scarcity of the literature, but also compared with the limit of integer case and good agreements are obtained. The proposed algorithm for the calculation of two-center nuclear attraction integrals over Slater type orbitals with noninteger principal quantum numbers in nonlined-up coordinate systems permits to avoid the interpolation procedure used to overcome the difficulty introduced by the presence of noninteger principal quantum numbers. Finally, numerical aspects of the presented formulae are analyzed under wide range of quantum numbers, orbital exponents and internuclear distances.

COMPUTATION OF MULTICENTER NUCLEAR-ATTRACTION INTEGRALS OF INTEGER AND NONINTEGER n SLATER ORBITALS USING AUXILIARY FUNCTIONS

A unified treatment of multicenter nuclear-attraction integrals with integer n and noninteger n* Slater-type orbitals (ISTOs and NISTOs) is described. Using different sets of series expansion formulas for NISTOs and their two-center distributions in terms of ISTOs at a displaced center obtained by one of the authors, the three-center nuclear-attraction integrals over NISTOs are expressed through the products of overlap and two-center nuclear-attraction integrals. The two-center overlap and nuclear-attraction integrals are calculated by the use of well-known auxiliary functions Aσ and Bk. Accuracy of the computer results is quite high for quantum numbers, screening constants, and location of orbitals.