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

Yıl 2022, Cilt: 17 Sayı: 2, 438 - 452, 25.11.2022
https://doi.org/10.29233/sdufeffd.1165376

Öz

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.

Kaynakça

  • T. Kato, “On the eigenfunctions of many-particle systems in quantum mechanics,” Comm. Pure Appl. Math., 10 (2), 151-177, 1957.
  • P. E. Hoggan, M. B. Ruiz, and T. Özdoğan, “Molecular Integrals over Slater-type orbitals. From pioneers to recent progress,” in Quantum Frontiers of Atoms and Molecules, M. V. Putz, Ed. New York: Nova Publishers Inc., 2010, pp. 63-90.
  • R. S. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, “Formulas and numerical tables for overlap integrals,” J. Chem. Phys., 17 (12), 1248-1267, 1949.
  • C. C. J. Roothaan, “A Study of two-center integrals useful in calculations on molecular structure. I,” J. Chem. Phys., 19 (12), 1445-1458, 1951.
  • K. Ruedenberg, “A study of two‐center integrals useful in calculations on molecular structure. II. The two‐center exchange integrals,” J. Chem. Phys., 19 (12), 1459-1477, 1951.
  • C. C. J. Roothaan, “Study of two‐center integrals useful in calculations on molecular structure. IV. The auxiliary functions Cαβγδε(ρa, ρb) for α≥0,” J. Chem. Phys., 24 (5), 947-960, 1956.
  • K. Ruedenberg, C. C. J. Roothaan, and W. Jaunzemis, “Study of two‐center integrals useful in calculations on molecular structure. III. A unified treatment of the hybrid, Coulomb, and one‐electron integrals,” J. Chem. Phys., 24 (2), 201-220, 1956.
  • A. C. Wahl, P. E. Cade, and C. C. J. Roothaan, “Study of two‐center integrals useful in calculations on molecular structure. V. General methods for diatomic integrals applicable to digital computers,” J. Chem. Phys., 41 (9), 2578-2599, 1964.
  • D. M. Silver and K. Ruedenberg, “Coulomb integrals between Slater-type atomic orbitals,” J. Chem. Phys., 49 (10), 4306-4311, 1968.
  • I. I. Guseinov, “Analytical evaluation of two-centre Coulomb, hybrid and one electron integrals for Slater type orbitals,” J. Phys. B, 3 (11), 1399-1412, 1970.
  • I. I. Guseinov, “Analytical evaluation of one‐ and two‐center Coulomb and two‐center hybrid integrals for Slater‐type orbitals,” J. Chem. Phys., 67 (8), 3837-3839, 1977.
  • J. Yasui and A. Saika, “Unified analytical evaluation of two‐center, two‐electron integrals over Slater‐type orbitals,” J. Chem. Phys., 76 (1), 468-472, 1982.
  • A. Özmen, A. Karakaş, Ü. Atav, and Y. Yakar, “Computation of two-center Coulomb integrals over Slater-type orbitals using elliptical coordinates,” Int. J. Quantum Chem., 91(1), 13-19, 2003.
  • M. P. Barnet and C. A. Coulson, “The evaluation of integrals occurring in the theory of molecular structure. Parts I & II,” Phil. Trans. R. Soc. Lond. A, 243 (864), 221-249, 1951.
  • P. O. Löwdin, “Quantum theory of cohesive properties of solids,” Adv. Phys., 5 (17), 1-171, 1956.
  • F. E. Harris and H. H. Michels, “Multicenter Integrals in Quantum Mechanics. I. Expansion of Slater-Type Orbitals about a New Origin,” J. Chem. Phys., 43 (10), 165-169, 1965.
  • W. England, “One-Center coulomb, two-center hybrid, and two-center Coulomb integrals over STP functions,” Int. J. Quantum Chem., 6 (3), 509-518, 1972.
  • R. R. Sharma, “Expansion of a function about a displaced center for multicenter integrals: A general and closed expression for the coefficients in the expansion of a Slater orbital and for overlap integrals,” Phys. Rev. A, 13 (2), 517-527, 1976.
  • H. W. Jones and C. A. Weatherford, “A modified form of Sharma's formula for sto Löwdin alpha functions with recurrence relations for the coefficient matrix,” Int. J. Quantum Chem. Symp., 14 (S12), 483-488, 1978.
  • H. W. Jones, “Computer-generated formulas for two-center coulomb integrals over Slater‒type orbitals,” Int. J. Quantum Chem., 20 (6), 1217-1224, 1981.
  • H. W. Jones, “Benchmark values for two-center Coulomb integrals over Slater-type orbitals,” Int. J. Quantum Chem., 45 (1), 21-30, 1993.
  • I. I. Guseinov, “Expansion of Slater-type orbitals about a displaced center and the evaluation of multicenter electron-repulsion integrals,” Phys Rev A, 31(5), 2851-2853, 1985.
  • I. I. Guseinov, “Unified analytical treatment of multicenter multielectron integrals of central and noncentral interaction potentials over Slater orbitals using 𝛹α-ETOs,” J. Chem. Phys., 119 (9), 4614-4619, 2003.
  • J. Fernandez Rico, R. Lopez, and G. Ramirez, “Calculation of integrals with Slater basis from the one-range expansion of the 0s function,” Int. J. Quantum Chem., 37 (1), 69-83, 1990.
  • V. Magnasco and A. Rapallo, “New translation method for STOs and its application to calculation of two-center two-electron integrals,” Int. J. Quantum Chem., 79 (2), 91-100, 2000.
  • M. Geller, “Two-Electron, one- and two-center Integrals,” J. Chem. Phys., 39 (3), 853-854, 1963.
  • M. Geller and R. W. Griffith, “Zero‐Field splitting, one‐and two‐center Coulomb‐type integrals,” J. Chem. Phys., 40 (8), 2309-2325, 1964.
  • M. Geller, “Two‐Center Coulomb integrals,” J. Chem. Phys., 41 (12), 4006-4007, 1964.
  • F. E. Harris, “Rapid evaluation of Coulomb integrals,” J. Chem. Phys., 51 (11), 4770-4778, 1969.
  • H. D. Todd, K. G. Kay, and H. J. Silverstone, “Unified treatment of two‐center overlap, Coulomb, and kinetic‐energy integrals,” J. Chem. Phys., 53 (10), 3951-3956, 1970.
  • E. Filter and E. O. Steinborn, “Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals,” Phys. Rev. A, 18 (1), 1-11, 1978.
  • H. P. Trivedi and E. O. Steinborn, “Fourier transform of a two-center product of exponential-type orbitals. Application to one-and two-electron multicenter integrals,” Phys. Rev. A, 27 (2), 670-679, 1983.
  • E. J. Weniger, J. Grotendorst, and E. O. Steinborn, “Unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions via the Fourier-transform method,” Phys. Rev. A, 33 (6), 3688-3705, 1986.
  • J. Grotendorst, E. J. Weniger, and E. O. Steinborn, “Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators,” Phys. Rev. A, 33 (6), 3706-3726, 1986.
  • J. Grotendorst and E. O. Steinborn, “Numerical evaluation of molecular one-and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method,” Phys. Rev. A, 38 (8), 3857-3876, 1988.
  • G. Figari, C. Costa, R. Pratolongo, and V. Magnasco, “Two-Centre Coulomb integrals over STOs from analytical evaluation of k-integrals by contour integration in the complex plane,” Chem. Phys. Lett., 167 (6), 547-554, 1990.
  • E. O. Steinborn, H. H. H. Homeier, and E. J. Weniger, “Recent progress on representations for Coulomb integrals of exponential-type orbitals,” J. Mol. Struct., 260, 207-221, 1992.
  • S. F. Boys, G. B. Cook, C. M. Reeves, and I. Shavitt, “Automatic fundamental calculations of molecular structure,” Nature, 178, 1207-1209, 1956.
  • J. Fernandez Rico, R. Lopez, A, Aguado, I. Ema, and G. Ramirez, “Reference program for molecular calculations with Slater-type orbitals,” J. Comp. Chem., 19 (11), 1284-1293, 1998.
  • I. Shavitt and M. Karplus, “Gaussian‐Transform method for molecular integrals. I. formulation for energy integrals,” J. Chem. Phys., 43 (2), 398-414, 1965.
  • J. Fernandez Rico, R. Lopez, I. Ema, and G. Ramirez, “Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules,” J. Comp. Chem., 25 (16), 1987-1994, 2004.
  • L. Berlu, H. Safohi, and P. E. Hoggan, “Fast and accurate evaluation of three-center, two-electron Coulomb, hybrid, and three-center nuclear attraction integrals over Slater-type orbitals using the SD transformation,” Int. J. Quantum Chem., 99 (4), 221-235, 2004.
  • H. Safohi and L. Berlu, “The Fourier transform method and the SD approach for the analytical and numerical treatment of multicenter overlap-like quantum similarity integrals,” J. Comp. Phys., 216 (1), 19-36, 2006.
  • S. Gümüş, “On the computation of Two-center Coulomb integrals over Slater type orbitals using the Poisson equation,” Z. Naturforsch A, 60a, 477-483, 2005.
  • P. E. Hoggan, “General two-electron exponential type orbital integrals in polyatomics without orbital translations,” Int. J. Quantum Chem., 109 (13), 2926-2932, 2009.
  • P. E. Hoggan, “Four-center Slater-type orbital molecular integrals without orbital translations,” Int. J. Quantum Chem., 110 (1), 98-103, 2010.
  • C. B. Mendl, “Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals,” J. Comp. Phys., 231 (15), 5157-5175, 2012.
  • M. Lesiuk and R. Moszynski, “Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals,” Phys. Rev. E, 90 (6), 063318, 2014.
  • A. Bağcı and P. E. Hoggan, “Benchmark values for molecular two-electron integrals arising from the Dirac equation,” Phys. Rev. E, 91 (2), 023303, 2015.
  • F. P. Prosser and C. H. Blanchard, “On the Evaluation of two‐center integrals,” J. Chem. Phys., 36 (4), 1112-1112, 1962.
  • S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addision Wesley, New York, 1998.
  • G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, London, 2005.
  • T. Özdoğan and M. Orbay, “Evaluation of two-center overlap and nuclear attraction integrals over slater-type orbitals with integer and noninteger principal quantum numbers,” Int. J. Quantum Chem., 87 (1), 15-22, 2002.
  • A. Bağcı and P. E. Hoggan, “Performance of numerical approximation on the calculation of overlap integrals with noninteger Slater-type orbitals,” Phys. Rev. E, 89 (5), 053307, 2014.
  • I. I. Guseinov and B. A. Mamedov, “On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals I. Single-center expansion method,” Int. J. Quantum Chem., 78 (3), 146-152, 2000.
  • E. Öztekin and S. Özcan, “Overlap integrals between irregular solid harmonics and STOs via the Fourier transform methods,” J. Math. Chem., 42 (3), 337-351, (2007).
  • I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, Academic Press, New York, 2000.
  • E. Öztekin, S. Özcan, M. Orbay, and M. Yavuz, “Calculation of nuclear-attraction and modified overlap integrals using Gegenbauer coefficients,” Int. J. Quantum Chem., 90 (1), 136-143, 2002.
  • S. A. Kurt, “Bazı atom ve moleküller için moleküler integrallerin hesaplanması,” M.S. thesis (Second Thesis Advisor: Selda Akdemir), Dept. Phys., Ondokuz Mayıs Univ., Samsun, Turkey, 2014.
  • J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw-Hill, New York, 1970.
  • I. I. Guseinov, B. A. Mamedov, and A. Rzaeva, “Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals,” J. Mol. Model., 8 (4), 145-149, 2002.
  • V. Magnasco, M. Casanova, and A. Rapallo, “On the evaluation of two-centre molecular integrals over an STO basis,” Chem. Phys. Lett., 289 (1-2), 81-89, (1998).

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

Yıl 2022, Cilt: 17 Sayı: 2, 438 - 452, 25.11.2022
https://doi.org/10.29233/sdufeffd.1165376

Öz

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.

Kaynakça

  • T. Kato, “On the eigenfunctions of many-particle systems in quantum mechanics,” Comm. Pure Appl. Math., 10 (2), 151-177, 1957.
  • P. E. Hoggan, M. B. Ruiz, and T. Özdoğan, “Molecular Integrals over Slater-type orbitals. From pioneers to recent progress,” in Quantum Frontiers of Atoms and Molecules, M. V. Putz, Ed. New York: Nova Publishers Inc., 2010, pp. 63-90.
  • R. S. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, “Formulas and numerical tables for overlap integrals,” J. Chem. Phys., 17 (12), 1248-1267, 1949.
  • C. C. J. Roothaan, “A Study of two-center integrals useful in calculations on molecular structure. I,” J. Chem. Phys., 19 (12), 1445-1458, 1951.
  • K. Ruedenberg, “A study of two‐center integrals useful in calculations on molecular structure. II. The two‐center exchange integrals,” J. Chem. Phys., 19 (12), 1459-1477, 1951.
  • C. C. J. Roothaan, “Study of two‐center integrals useful in calculations on molecular structure. IV. The auxiliary functions Cαβγδε(ρa, ρb) for α≥0,” J. Chem. Phys., 24 (5), 947-960, 1956.
  • K. Ruedenberg, C. C. J. Roothaan, and W. Jaunzemis, “Study of two‐center integrals useful in calculations on molecular structure. III. A unified treatment of the hybrid, Coulomb, and one‐electron integrals,” J. Chem. Phys., 24 (2), 201-220, 1956.
  • A. C. Wahl, P. E. Cade, and C. C. J. Roothaan, “Study of two‐center integrals useful in calculations on molecular structure. V. General methods for diatomic integrals applicable to digital computers,” J. Chem. Phys., 41 (9), 2578-2599, 1964.
  • D. M. Silver and K. Ruedenberg, “Coulomb integrals between Slater-type atomic orbitals,” J. Chem. Phys., 49 (10), 4306-4311, 1968.
  • I. I. Guseinov, “Analytical evaluation of two-centre Coulomb, hybrid and one electron integrals for Slater type orbitals,” J. Phys. B, 3 (11), 1399-1412, 1970.
  • I. I. Guseinov, “Analytical evaluation of one‐ and two‐center Coulomb and two‐center hybrid integrals for Slater‐type orbitals,” J. Chem. Phys., 67 (8), 3837-3839, 1977.
  • J. Yasui and A. Saika, “Unified analytical evaluation of two‐center, two‐electron integrals over Slater‐type orbitals,” J. Chem. Phys., 76 (1), 468-472, 1982.
  • A. Özmen, A. Karakaş, Ü. Atav, and Y. Yakar, “Computation of two-center Coulomb integrals over Slater-type orbitals using elliptical coordinates,” Int. J. Quantum Chem., 91(1), 13-19, 2003.
  • M. P. Barnet and C. A. Coulson, “The evaluation of integrals occurring in the theory of molecular structure. Parts I & II,” Phil. Trans. R. Soc. Lond. A, 243 (864), 221-249, 1951.
  • P. O. Löwdin, “Quantum theory of cohesive properties of solids,” Adv. Phys., 5 (17), 1-171, 1956.
  • F. E. Harris and H. H. Michels, “Multicenter Integrals in Quantum Mechanics. I. Expansion of Slater-Type Orbitals about a New Origin,” J. Chem. Phys., 43 (10), 165-169, 1965.
  • W. England, “One-Center coulomb, two-center hybrid, and two-center Coulomb integrals over STP functions,” Int. J. Quantum Chem., 6 (3), 509-518, 1972.
  • R. R. Sharma, “Expansion of a function about a displaced center for multicenter integrals: A general and closed expression for the coefficients in the expansion of a Slater orbital and for overlap integrals,” Phys. Rev. A, 13 (2), 517-527, 1976.
  • H. W. Jones and C. A. Weatherford, “A modified form of Sharma's formula for sto Löwdin alpha functions with recurrence relations for the coefficient matrix,” Int. J. Quantum Chem. Symp., 14 (S12), 483-488, 1978.
  • H. W. Jones, “Computer-generated formulas for two-center coulomb integrals over Slater‒type orbitals,” Int. J. Quantum Chem., 20 (6), 1217-1224, 1981.
  • H. W. Jones, “Benchmark values for two-center Coulomb integrals over Slater-type orbitals,” Int. J. Quantum Chem., 45 (1), 21-30, 1993.
  • I. I. Guseinov, “Expansion of Slater-type orbitals about a displaced center and the evaluation of multicenter electron-repulsion integrals,” Phys Rev A, 31(5), 2851-2853, 1985.
  • I. I. Guseinov, “Unified analytical treatment of multicenter multielectron integrals of central and noncentral interaction potentials over Slater orbitals using 𝛹α-ETOs,” J. Chem. Phys., 119 (9), 4614-4619, 2003.
  • J. Fernandez Rico, R. Lopez, and G. Ramirez, “Calculation of integrals with Slater basis from the one-range expansion of the 0s function,” Int. J. Quantum Chem., 37 (1), 69-83, 1990.
  • V. Magnasco and A. Rapallo, “New translation method for STOs and its application to calculation of two-center two-electron integrals,” Int. J. Quantum Chem., 79 (2), 91-100, 2000.
  • M. Geller, “Two-Electron, one- and two-center Integrals,” J. Chem. Phys., 39 (3), 853-854, 1963.
  • M. Geller and R. W. Griffith, “Zero‐Field splitting, one‐and two‐center Coulomb‐type integrals,” J. Chem. Phys., 40 (8), 2309-2325, 1964.
  • M. Geller, “Two‐Center Coulomb integrals,” J. Chem. Phys., 41 (12), 4006-4007, 1964.
  • F. E. Harris, “Rapid evaluation of Coulomb integrals,” J. Chem. Phys., 51 (11), 4770-4778, 1969.
  • H. D. Todd, K. G. Kay, and H. J. Silverstone, “Unified treatment of two‐center overlap, Coulomb, and kinetic‐energy integrals,” J. Chem. Phys., 53 (10), 3951-3956, 1970.
  • E. Filter and E. O. Steinborn, “Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals,” Phys. Rev. A, 18 (1), 1-11, 1978.
  • H. P. Trivedi and E. O. Steinborn, “Fourier transform of a two-center product of exponential-type orbitals. Application to one-and two-electron multicenter integrals,” Phys. Rev. A, 27 (2), 670-679, 1983.
  • E. J. Weniger, J. Grotendorst, and E. O. Steinborn, “Unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions via the Fourier-transform method,” Phys. Rev. A, 33 (6), 3688-3705, 1986.
  • J. Grotendorst, E. J. Weniger, and E. O. Steinborn, “Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators,” Phys. Rev. A, 33 (6), 3706-3726, 1986.
  • J. Grotendorst and E. O. Steinborn, “Numerical evaluation of molecular one-and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method,” Phys. Rev. A, 38 (8), 3857-3876, 1988.
  • G. Figari, C. Costa, R. Pratolongo, and V. Magnasco, “Two-Centre Coulomb integrals over STOs from analytical evaluation of k-integrals by contour integration in the complex plane,” Chem. Phys. Lett., 167 (6), 547-554, 1990.
  • E. O. Steinborn, H. H. H. Homeier, and E. J. Weniger, “Recent progress on representations for Coulomb integrals of exponential-type orbitals,” J. Mol. Struct., 260, 207-221, 1992.
  • S. F. Boys, G. B. Cook, C. M. Reeves, and I. Shavitt, “Automatic fundamental calculations of molecular structure,” Nature, 178, 1207-1209, 1956.
  • J. Fernandez Rico, R. Lopez, A, Aguado, I. Ema, and G. Ramirez, “Reference program for molecular calculations with Slater-type orbitals,” J. Comp. Chem., 19 (11), 1284-1293, 1998.
  • I. Shavitt and M. Karplus, “Gaussian‐Transform method for molecular integrals. I. formulation for energy integrals,” J. Chem. Phys., 43 (2), 398-414, 1965.
  • J. Fernandez Rico, R. Lopez, I. Ema, and G. Ramirez, “Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules,” J. Comp. Chem., 25 (16), 1987-1994, 2004.
  • L. Berlu, H. Safohi, and P. E. Hoggan, “Fast and accurate evaluation of three-center, two-electron Coulomb, hybrid, and three-center nuclear attraction integrals over Slater-type orbitals using the SD transformation,” Int. J. Quantum Chem., 99 (4), 221-235, 2004.
  • H. Safohi and L. Berlu, “The Fourier transform method and the SD approach for the analytical and numerical treatment of multicenter overlap-like quantum similarity integrals,” J. Comp. Phys., 216 (1), 19-36, 2006.
  • S. Gümüş, “On the computation of Two-center Coulomb integrals over Slater type orbitals using the Poisson equation,” Z. Naturforsch A, 60a, 477-483, 2005.
  • P. E. Hoggan, “General two-electron exponential type orbital integrals in polyatomics without orbital translations,” Int. J. Quantum Chem., 109 (13), 2926-2932, 2009.
  • P. E. Hoggan, “Four-center Slater-type orbital molecular integrals without orbital translations,” Int. J. Quantum Chem., 110 (1), 98-103, 2010.
  • C. B. Mendl, “Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals,” J. Comp. Phys., 231 (15), 5157-5175, 2012.
  • M. Lesiuk and R. Moszynski, “Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals,” Phys. Rev. E, 90 (6), 063318, 2014.
  • A. Bağcı and P. E. Hoggan, “Benchmark values for molecular two-electron integrals arising from the Dirac equation,” Phys. Rev. E, 91 (2), 023303, 2015.
  • F. P. Prosser and C. H. Blanchard, “On the Evaluation of two‐center integrals,” J. Chem. Phys., 36 (4), 1112-1112, 1962.
  • S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addision Wesley, New York, 1998.
  • G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, London, 2005.
  • T. Özdoğan and M. Orbay, “Evaluation of two-center overlap and nuclear attraction integrals over slater-type orbitals with integer and noninteger principal quantum numbers,” Int. J. Quantum Chem., 87 (1), 15-22, 2002.
  • A. Bağcı and P. E. Hoggan, “Performance of numerical approximation on the calculation of overlap integrals with noninteger Slater-type orbitals,” Phys. Rev. E, 89 (5), 053307, 2014.
  • I. I. Guseinov and B. A. Mamedov, “On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals I. Single-center expansion method,” Int. J. Quantum Chem., 78 (3), 146-152, 2000.
  • E. Öztekin and S. Özcan, “Overlap integrals between irregular solid harmonics and STOs via the Fourier transform methods,” J. Math. Chem., 42 (3), 337-351, (2007).
  • I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, Academic Press, New York, 2000.
  • E. Öztekin, S. Özcan, M. Orbay, and M. Yavuz, “Calculation of nuclear-attraction and modified overlap integrals using Gegenbauer coefficients,” Int. J. Quantum Chem., 90 (1), 136-143, 2002.
  • S. A. Kurt, “Bazı atom ve moleküller için moleküler integrallerin hesaplanması,” M.S. thesis (Second Thesis Advisor: Selda Akdemir), Dept. Phys., Ondokuz Mayıs Univ., Samsun, Turkey, 2014.
  • J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw-Hill, New York, 1970.
  • I. I. Guseinov, B. A. Mamedov, and A. Rzaeva, “Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals,” J. Mol. Model., 8 (4), 145-149, 2002.
  • V. Magnasco, M. Casanova, and A. Rapallo, “On the evaluation of two-centre molecular integrals over an STO basis,” Chem. Phys. Lett., 289 (1-2), 81-89, (1998).
Toplam 62 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Metroloji,Uygulamalı ve Endüstriyel Fizik
Bölüm Makaleler
Yazarlar

Selda Akdemir 0000-0002-5487-8703

Yayımlanma Tarihi 25 Kasım 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 17 Sayı: 2

Kaynak Göster

IEEE S. Akdemir, “Properties of One- and Two-Center Coulomb Integrals over Slater Type Orbitals”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, c. 17, sy. 2, ss. 438–452, 2022, doi: 10.29233/sdufeffd.1165376.