Chapter 9 - Reducing and Solving Electric Multipole Moment Integrals

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

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.

In this study, two-electron one- and two-center Coulomb integrals with the same and different screening parameters are investigated numerically in the real Slater type orbital (STO) basis using Fourier transform method. In momentum space firstly, for atomic, i.e. one-center, Coulomb integrals are calculated, and analytical expressions are obtained in terms of binomial coefficients. Then, the solutions of the two-center Coulomb integrals are made with the modified Bessel function of second kind and the results are expressed in terms of binomial and Gaunt coefficients, irregular solid harmonics, and finite sum of STOs. A computer program is written in the MATHEMATICA language to determine the accuracy of the analytical expressions that are highly suitable for programming. The numerical results obtained from the program are given in the tables, and it is shown that the results agree with the literature.

AbstractÖzdoǧan (J. Chin. Chem. Soc. 2004, 51, 13) published formulas for evaluating the multicenter electric multipole moment (EMM) integrals over Slater‐type orbitals (STOs). It is demonstrated that all of the formulas presented in this work are not original and they are available in our already published papers or can easily be derived by means of a simple algebra (I. I. Guseinov, J. Mol. Struct. (Theochem) 1996, 367, 83; 1998, 427, 263; Int. J. Quantum Chem. 1998, 68, 145; I. I. Guseinov, B. A. Mamedov, Int. J. Quantum Chem. 2003, 93, 9.

Computation of molecular integrals over Slater type orbitals. II. Calculation of electric multipole moment integrals using translation formulas

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

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.

Polynomials for evaluation of two-center overlap integrals

The analytical expressions are derived for the magnetic multipole moment integrals in terms of electric multipole moment integrals for which the closed formulas through the overlap integrals are obtained. By the use of the derived expressions in terms of overlap integrals, the electric and magnetic multipole moment integrals, the electric and magnetic properties of molecules can be evaluated most efficiently and accurately. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 145–150, 1998

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function \({G_{-ns}^q }\) which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function \({G_{-ns}^q (p_a ,p,pt)}\) is completely general for p a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function \({G_{-ns}^q}\) is compared with other method.

Despite numerous potential advantages of subradiative dark modes due to an extremely high quality ( ${Q}$ ) factor, the practicality still depends on the consistency of excitations for a tight fabrication tolerance. This article proposes a novel approach to adaptive transitions between different dipole moments to allow for consistent excitation of Fano and electric quadrupole resonances that are extremely sensitive to geometric errors. The proposed geometry of each metamolecule consists of four strips printed on a thin dielectric film with four gaps between the strips, and the geometric error is described as structural deformation by adjusting the strip lengths and gap sizes in the vertical direction. In each deformed state, current and magnetic-field distributions are analyzed to interpret operating principles, and multipole moments are calculated based on the multipole expansion using the current distributions. The results demonstrate that the consistent excitation of the electric quadrupole moment is enabled based on a simultaneous transition between toroidal and electric dipole moments, and the Fano resonance is consistently excited by the magnetic-to-electric dipole moment transition. For verification, original and deformed metasurfaces are fabricated, and their transmission and reflection coefficients are measured in a semianechoic chamber to experimentally validate the proposed dipole moment transitions. The results confirm that the proposed approach has great potential to bring advances in recent sensing and energy harvesting applications where an extremely high ${Q}$ -factor is essential.

A program written in the MAPLE symbolic computational language was written to compute analytic and numerical expressions for the long-range multipole moment expansion of the electrostatic (Coulombic) perturbation operator for a system of N atoms in arbitrary orientations in space. These expressions are written in terms of the multipole moment operators on individual interacting atoms. The program was tested on a system consisting of a ground-state hydrogen atom interacting with two protons in three different configurations, and the results were found to be in agreement with manual calculations. This approach can produce, with relative ease, the long-range multipole expansion for systems with large numbers of interacting atoms or ions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 343–351, 1997

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

Ab initio calculations of electric multipole moments, (higher) polarizabilities and first hyperpolarizabilitiy of (H2S)n, n = 1 – 4

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.