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    Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators
    (2023) Akdemir, Selda
    In this study, Gaunt coefficients, frequently encountered in quantum mechanical calculations of atomic and molecular structures, have been algebraically derived. Firstly, the Gaunt coefficient, equal to the integral over the solid angle of the product of three spherical harmonics, is written in terms of angular momentum ladder operators. Subsequently, raising or lowering operators are applied to spherical harmonics, and the obtained integrals are solved using the recurrence and orthogonality relations of spherical harmonics. As a result, algebraic expressions for Gaunt coefficients are obtained in terms of quantum numbers
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    Convergence of Slater-Type Orbitals in Calculations of Basic Molecular Integrals
    (Springer International Publishing Ag, 2018) Akdemir, Selda
    The relationship between real Slater-type orbitals with the distinct scaling constants is examined analytically via the Fourier transform method. The convergence of the formula that we have derived in terms of infinite sums of Slater-type orbitals is analyzed numerically. Subsequently, the analytical expression is applied to basic molecular integrals. Numerical calculations performed to demonstrate the accuracy of the obtained formulas are compared with results in the literature. Numerical results are also presented in tables.
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    Evaluation of One-Electron Basic Integrals of Irregular Solid Harmonics and Slater-Type Orbitals Using Fourier Transforms
    (Elsevier Academic Press Inc, 2016) Akdemir, Selda; Yukcu, Niyazi; Oztekin, Emin
    In this paper, for one-and two-center one-electron integrals between irregular solid harmonics (ISHs) and Slater-type orbitals (STOs) with same screening parameters distinct analytical formulas are obtained. First, an expression is derived in terms of ISHs and STOs using a Fourier transform method and Taylor expansion. The second expression is written by taking the two-center overlap integrals over STOs with the different screening parameters to a limit. Numerical calculations of these two formulae have been performed for chosen quantum numbers. Finally, the two-center basic nuclear attraction integrals have been calculated as a special case of these integrals between ISHs and STOs. Results obtained have been compared with the literature. This comparison shows a perfect match.
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    Properties of One- and Two-Center Coulomb Integrals over Slater Type Orbitals
    (2022) Akdemir, Selda
    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.