Properties of One- and Two-Center Coulomb Integrals over Slater Type Orbitals

Year 2022, Volume: 17 Issue: 2, 438 - 452, 25.11.2022

Selda Akdemir

https://doi.org/10.29233/sdufeffd.1165376

Abstract

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.

Keywords

Coulomb integral, Fourier transform method, Slater type atomic orbital, Taylor expansion

Slater Tipi Orbitaller Bazında Bir- ve İki-Merkezli Coulomb İntegrallerinin Özellikleri

Abstract

Bu çalışmada, aynı ve farklı perdeleme sabitlerine sahip iki elektronlu bir- ve iki-merkezli Coulomb integralleri, Fourier dönüşüm yöntemi kullanılarak reel Slater tipi orbitaller (STO) bazında sayısal olarak incelenmiştir. Momentum uzayında ilk olarak atomik, yani tek-merkezli, Coulomb integralleri için hesaplama yapılmış ve analitik ifadeler binom katsayıları cinsinden elde edilmiştir. Daha sonra, iki-merkezli Coulomb integrallerinin çözümleri, ikinci tür modifiye edilmiş Bessel fonksiyonları ile yapılmış ve sonuçlar binom ve Gaunt katsayıları, düzensiz katı harmonikler ve STO’ların sonlu toplamı cinsinden ifade edilmiştir. Programlamaya son derece uygun olan analitik ifadelerin doğruluğunu belirlemek için MATHEMATICA dilinde bir bilgisayar programı yazılmıştır. Programdan elde edilen sayısal sonuçlar tablolarda verilmiş ve sonuçların literatür ile uyumlu olduğu gösterilmiştir.

Keywords

Coulomb integrali, Fourier dönüşüm metodu, Slater tip atomik orbital, Taylor açılımı.

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