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Öğe Asymptotic behavior of Clebsch-Gordan coefficients(Springer, 2024) Akdemir, S.; Ozay, S.; Oztekin, E.In this study, the Clebsch-Gordan coefficients are expressed using the generalized hypergeometric functions of unit argument and binomial coefficients, and their behavior as a function of quantum numbers is examined. The Clebsch-Gordan coefficients are also numerically computed for integer and non-integer quantum numbers. Subsequently, the convergence between the Clebsch-Gordan and asymptotic Clebsch-Gordan coefficients is investigated based on the orbital angular momentum quantum numbers.Öğe Calculation of electric multipole moment integrals with the different screening parameters via the Fourier transform method(Wiley, 2012) Yukcu, N.; Senlik, I.; Oztekin, E.Three-center electric multipole moment integrals over Slater-type orbitals (STOs) can be evaluated by translating the orbitals on one center to the other and reducing the system to an expansion of two-center integrals. These are then evaluated using Fourier transforms. The resulting expression depends on the overlap integrals that can be evaluated with the greatest ease. They involve expressions for STO with different screening parameters that are known analytically. This work gives the overall expressions analytically in a compact form, based on Gegenbauer polynomials. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem 112: 414-425, 2012Öğe Generating function for rotation matrix elements(Wiley-Blackwell, 2012) Akdemir, S. O.; Oztekin, E.The rotation matrix elements are expressed in terms of the Jacobi, Hypergeometric, and Legendre polynomials in the literature. In this study, the generating function is presented for rotation matrix elements by using properties of Jacobi polynomials. In addition, some special values and Rodrigues' formula of rotation matrix elements are obtained by using the generating function. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012Öğe New analytical expressions, symmetry relations and numerical solutions for the rotational overlap integrals(Wiley, 2012) Akdemir, S. O.; Eryilmaz, S. D.; Oztekin, E.In this article, extremely simple analytical formulas are obtained for rotational overlap integrals which occur in integrals over two reduced rotation matrix elements. The analytical derivations are based on the properties of the Jacobi polynomials and beta functions. Numerical results and special values for rotational overlap integrals are obtained by using symmetry properties and recurrence relationships for reduced rotation matrix elements. The final results are of surprisingly simple structures and very useful for practical applications. (c) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012Öğe New orthogonality relationships of the Gaunt coefficients(Elsevier, 2024) Ozay, S.; Akdemir, S.; Oztekin, E.A new analytical formula for the Gaunt coefficients is derived using expressions of Clebsch Gordan coefficients in terms of the generalized hypergeometric functions of unit argument and binomial coefficients. In addition, new sum expressions and orthogonality relations containing the Gaunt coefficients are written. These expressions are used to test the accuracy of the numerical calculation for the Gaunt coefficients. These formulae are coded in the Mathematica programming language, and the Gaunt coefficients are calculated numerically and compared with the values from the literature.Öğe Reducing and Solving Electric Multipole Moment Integrals(Elsevier Academic Press Inc, 2013) Yukcu, N.; Oztekin, E.Electric multipole moment integrals with the same screening parameters are calculated using Slater type orbitals (STOs) translation formulae and the Fourier transform method. First, multipole moment operators which appear in the three-center electric multipole moment integrals are translated from the 0-center to the b-center. After the translation procedure and using Fourier transform convolution theorem, three-center electric multipole moment integrals were reduced to two-center molecular integrals. Then, the analytical expressions obtained were written in terms of overlap integrals. The numerical values for electric multiple moment integrals are useful especially for high quantum numbers and atomic parameters.Öğe The coupling coefficients with six parameters and the generalized hypergeometric functions(Elsevier, 2025) Ozay, S.; Akdemir, S.; Oztekin, E.In this study, the Gaunt coefficients, Clebsch-Gordan coefficients, and the Wigner 3j and 6j symbols are expressed as the product of generalized hypergeometric functions with unit argument and a normalization coefficient. By exploiting the symmetry properties of generalized hypergeometric functions, these functions are transformed into numerically computable forms, and the normalization coefficients are fully expressed in terms of binomial coefficients. New mathematical expressions, in the form of a series of products of three Gaunt coefficients, are presented, which can be used to verify the accuracy of numerical calculations. An algorithm has been developed to compute binomial coefficients and generalized hypergeometric functions using recurrence relations, eliminating the need for factorial functions. Utilizing this algorithm and the derived analytical expressions, the Gaunt_CG_3j_and_6j Mathematica program, which numerically calculates the Gaunt coefficients, Clebsch-Gordan coefficients, and the Wigner 3j and 6j symbols, was written without relying on Mathematica's built-in functions. The program can be easily adapted to other programming languages and run on all versions of Mathematica. Program Summary: Program title: Gaunt_CG_3j_and_6j CPC Library link to program files: https://doi.org/10.17632/pwhry4278g.1 Licensing provisions: GPLv2 Programming language: Wolfram Language (Mathematica 9.0 or higher) Nature of problem: In this study, analytical expressions for the Gaunt coefficients and Wigner 6j symbols are derived as the product of generalized hypergeometric functions and a normalization coefficient. Analytical expressions for the Wigner 3j symbols in terms of the Clebsch-Gordan (CG) coefficients are given in the same form in Ref. [27]. This study introduces new analytical expressions involving sums to verify the accuracy of numerical calculations for all coupling coefficients with six parameters consolidated into a single formula. The Gaunt_CG_3j_and_6j program numerically calculates the coupling coefficients presented in this study and Ref. [27]. Solution method: In this study, all coupling coefficients with six parameters are expressed as the product of the generalized hypergeometric function with unit argument and the normalization constant, which is written in terms of binomial coefficients. Therefore, the main structure of our program consists of numerical calculation of the binomial coefficients and generalized hypergeometric functions with unit argument. In our algorithm, binomial coefficients with parameters containing negative or positive integers are calculated using the binomcag and binomcalc modules, based on Eq. (12) in Ref. [31]. The binomcalc module computes the binomial coefficients using the recurrence relation: n F(n, m) = n -m F(n - 1, m), n >= 1, m >= 0 with the initial conditions F(m, m) = 1 and F(n, m) = 0, form > n. To calculate the coupling coefficients with six parameters, generalized hypergeometric functions and binomial coefficients can be used, as demonstrated in this study, as an alternative to recurrence relations or serial ex-pressions and factorial functions. We implemented the hypergeowhl and hipergeomcal modules in our algorithm to calculate the generalized hypergeometric functions. We used Eq. (21) in Ref. [29] to compute the generalized hypergeometric functions with unit argument. Additional comments including Restrictions and Unusual features: When determining CPU times, issues unrelated to our program arise from Mathematica. The command lines we use to measure CPU time are: x1=TimeUsed[]; 3 vertical dots x2= TimeUsed[]; time=x2-x1 We encountered two problems when determining CPU times in this manner. The first problem occurs when we try to measure the calculation time for a single coupling coefficient, where Mathematica reports this time as zero (in seconds). We calculated the average CPU time by computing 1000 or more coupling coefficients for a single quantum set to overcome this. The second problem arises from the CPU times for non-zero coupling coefficients (particularly in expressions involving sums), which are unstable and vary each time the program is run. Since these results are reproducible across multiple program executions, we take the average of the computed times. Long write-up
