Rapid Evaluation of Coulomb Integrals

The present work is dedicated to the introduction of a new single‐center expansion method derived from the Barnett and Coulson approach leading to reliable numerical algorithms. The basic concept lies in the use of the integral representations of the modified Bessel functions Iλ+1/2 and Kλ+1/2, rather than employing their closed analytical expressions that are known to involve large numbers that usually make the numerical calculation worthless. It is applied to two‐center overlap and Coulomb integrals as well as to three‐center nuclear attraction integrals useful in density functional theory. © 1994 John Wiley & Sons, Inc.

In electronic structure theory, the charge distribution of a nucleus is usually approximated by point charge, Gaussian function, or homogeneously charged sphere, because they have an analytical nuclear attraction integral (NAI) formula. However, these functions do not always provide good approximations for nuclei with large mass number. The two-parameter Fermi (2pF) distribution and more realistic distributions describe well even nuclei with large mass number but do not have analytical NAI formulas. We propose a new function model called augmented Gaussian 12 (AG12), which has sufficient number of parameters and analytical NAI formulas. With the proposed fitting scheme, the AG12 charge distribution model optimally reproduces 2pF and the more realistic charge distributions. Moreover, AG12 fitted to 2pF model reproduces the energy difference of hydrogen-like ions well between the Gaussian distribution and 2pF models. Calculations using AG12 also suggested necessity to use more realistic nuclear charge distributions than 2pF.

Based on the property of two functions Snlmn′l′m′(R) and Wnlmn′l′m′(R) introduced in the Fock representation of the momentum-space Schrödinger equation for one-electron diatomics, we derive various relations and sum rules for the one-, two-, and three-center nuclear attraction integrals over (bound state) hydrogenic orbitals with a common exponent.

We have investigated the applicability of a molecular orbital method for co-ordination compounds, based on Mulliken-type approximations, using TiF63– as a control system. The method requires empirical evaluation of one-centre parts of the matrix elements and theoretical calculation of overlap, kinetic, Coulomb, and exchange two-centre integrals and of three-centre nuclear attraction integrals. Open-shell characteristics and electronic repulsion corrections for transition energies are taken into account. The use of several fixed configurations, or of a variable configuration, for the atomic eigenfunctions is tested. The calculations indicate a large participation of 4s and 4p orbitals to a very covalent bond, although a relatively small amount of π-bonding is present. However, the occupation numbers and some transition energies often become unacceptable. There appears to be evidence that this may be due to the lack of correlation for the two-centre interactions. In fact, all the variations that act in the same sense of a two-centre correlation tend to improve the results.

Proceeding from analytical expressions for two‐center kernel functions that we derived recently, we present new analytical formulas for the two‐center Coulomb and overlap integrals over Slater‐type orbitals. These formulas are of an exceptionally simple analytical structure and high numerical efficiency. An especially important point is that for the most frequently needed ranges of discrete quantum numbers, the formulas are completely stable in the cases of nearly equal scaling parameters or vanishing interatomic distances, except for one particular case of the Coulomb integral. No special asymptotic formulas are needed any more to compute the two‐center integrals over Slater‐type orbitals in these case. Furthermore, a largely recursive formulation makes the integral evaluation very economical and fast. In particular, we assess the numerical performance of a new kind of angular momentum recurrences that we have proposed in a previous article [W. Hierse and P.M. Oppeneer, J. Chem. Phys. 99, 1278 (1993)]. © 1994 John Wiley & Sons, Inc.

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ1aug and ℐ2aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature.

New formula for integrals in semiempirical molecular orbital methods: Part 1. General theory and results for two-center overlap integrals and their first and second derivatives with respect to Cartesian nuclear displacements

Two-centre overlap integrals involving mixed Slater-gaussian orbitals

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

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.

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).

The relationship between the wavefunctions for mostly 3d molecular orbitals and the energies of the nearly degenerate A41g and E4g excited states of octahedral Mn2+ complexes is investigated from two points of view. First, the expressions relating the energy splitting to the degree of covalency have been extended within the framework of the intermediate neglect of differential overlap (INDO) approximation to include the large but previously neglected two-center Coulomb integrals and all ligand one-center Coulomb and exchange integrals. The results resemble those previously obtained in their inability to account for the spectral assignment of the E4g state below E41g for the MnF64− complex in the salts KMnF3, RbMnF3, and MnF2. Second, the assumption of identical metal-ion repulsion integrals for the eg and t2g molecular orbitals is discarded in a simple analysis of an all-electron molecular SCF calculation for the MnF64− complex. While both the eg and t2g orbitals contain a metal-ion d orbital that is expanded relative to that for the free Mn2+ ion, the slightly greater radial expansion for the t2g orbital is shown to qualitatively account for the observed ordering of the A41g and E4g states.

Quantum mechanics of diatomic molecules: overlap integrals, coulomb integrals and ab initio calculations on imidogen

Density functional theory requires precise numerical values for three-center nuclear attraction integrals, best obtained over Slater-type orbitals (STOs). Efficient evaluation of three-center nuclear attraction integrals over STOs to predetermined accuracy is made possible by applying the nonlinear D and D transformations. These methods are implemented in Fortran subroutines. This work shows how the conditions of applicability for these transformations are readily proven to be satisfied using the Axiom symbolic computation system. Axiom also provides exact values of the integrals for comparison. Their evaluation by standard numerical quadrature methods (Gauss–Laguerre), proves inadequate: Certain parameters lead to inaccurate values for the integrals, whereas the transformed integrals are highly accurate and rapidly evaluated. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 181–188, 1998

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