Reducing and Solving Electric Multipole Moment Integrals

Yükçü, Niyazi; Öztekin, Emin

doi:10.1016/b978-0-12-411544-6.00009-1

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…It's well known that overlap integrals are the most basic molecular integrals and all other multi-center molecular integrals can be expressed in terms of them. For example, following electric multipole moment integrals are given in terms of overlap integrals [27]:…”

Section:   mentioning

confidence: 99%

“…In computational physics, this situation provides great advantage in molecular integral calculations. Because, the correct calculation of each of these coefficients and functions is important for accurate numerical results, and some of these works can be found in the references: [19][20][21][22][23][24][25][26][27]. The evaluation of definite integrals appears repeatedly in problems of mathematical physics as well as in pure mathematics.…”

Section: Introductionmentioning

confidence: 99%

“…. When 1 p  and 1 q  , the beta function is obtained from Equation (27). The another is 1 p  and 0 q  situation, and then the value of beta function can be found from   1, Bq .…”

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confidence: 99%

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The Numerical Evaluation Methods for Beta Function

Yükçü¹

2022

Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi

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In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented.

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Section:   mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Numerical Evaluation Methods for Beta Function

Yükçü¹

2022

Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi

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Analytical treatments for magnetic multipole moment integrals

Yükçü

Öztekin

2021

Chemical Physics Letters

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Calculation of the Gaunt coefficients over real spherical harmonics

Yükçü,

Atalay,

Yükçü

et al. 2024

Can. J. Phys.

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The Gaunt coefficients are numerical coefficients that describe the interaction between two different angular momentum quantum states. In the quantum physics, this interaction affects the energy levels and structures of atoms and molecules. The calculation of Gaunt coefficients play a significant role in molecular integral evaluations which emerge from the solving Schrödinger equation. This work presents the analytical evaluation methods for the Gaunt coefficients using real spherical harmonics and associated Legendre functions. In this paper, the general integral form of the Gaunt coefficients is written in two main parts as polar and azimuthal integrals. The new analytical relations and symmetry properties are generated for these integrals. The analytical expressions are obtained in terms of the beta functions, and these coefficients are calculated for different atomic orbitals of angular momentum using the Mathematica programming language. The obtained numerical results are compared with the values in the literature and we found completely agreement to 15 digits.

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Reducing and Solving Electric Multipole Moment Integrals

Cited by 3 publications

References 27 publications

The Numerical Evaluation Methods for Beta Function

The Numerical Evaluation Methods for Beta Function

Analytical treatments for magnetic multipole moment integrals

Calculation of the Gaunt coefficients over real spherical harmonics

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