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İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller

Year 2023, Volume: 9 Issue: 2, 375 - 384, 30.06.2023

Meral Coşkun , Murat Ertürk

https://doi.org/10.28979/jarnas.1163388

Abstract

Bu çalışmanın amacı, baş kuantum sayısı tamsayı olmayan Bessel tipli orbitallerin Hartree-Fock-Roothaan yöntemi ile atomik sistemlere uygulanabilirliğini ve literatürdeki diğer üstel tipli orbitallerden üstünlüklerini incelemektir. Birleşik Hartree-Fock-Roothaan yönteminde yeni önerilen Bessel tipli orbitaller kullanılarak, iki elektronlu atomik sistemlerin orbital ve temel durum enerji değerleri hesaplanmıştır. Minimal baz çerçevesinde oluşturulan yeni baz fonksiyonları ile elde edilen değerler tablolarda karşılaştırmalı olarak verilmiştir. Elde edilen sonuçlar, literatürde kullanılan benzer üstel tipli baz fonksiyonlarına göre daha iyi değerler vermekte ve sayısal Hartree-Fock değerleri ile çok iyi uyum sağlamaktadır.

Keywords

Baz fonksiyonu bessel fonksiyonu hartree-fock-roothaan yöntemi izoelektronik seriler kesir değerli baş kuantum sayısı

Supporting Institution

Çanakkale Onsekiz Mart Üniversitesi Bilimsel Araştırma Projeleri Koordinasyon Birimi

Project Number

FYL-2019-2851

Thanks

Bu çalışma, Çanakkale Onsekiz Mart Üniversitesi Bilimsel Araştırma Projeleri Koordinasyon Birimi tarafından FYL-2019-2851 nolu proje ile desteklenmiştir.

References

  • Referans1 Agmon, S. (1982). Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations: Bound on Eigen functions of N-Body Schrödinger Operators. Princeton University Press, Princeton. Erişim adresi: https://press.princeton.edu/books/hardcover/9780691641423/lectures-on-exponential-decay-of-solutions-of-second-order-elliptic
  • Referans2 Allouche, A. (1974). Les orbitales de Slater à nombre quantique ≪n≫ non-entier. Theor. Chim. Acta34(1), 79-83. DOI: https://doi.org/10.1007/BF00553235
  • Referans3 Bishop, D. M. ve Leclerc, J. C. (1972). Unconventional basis sets in quantum mechanical calculations. Mol. Phys., 24(5), 979-992. DOI: https://doi.org/10.1080/00268977200102091
  • Referans4 Boys, S. F. (1950). Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system. Proc. Roy. Soc. London A., 200(1063), 542-554. DOI: https://doi.org/10.1098/rspa.1950.0036
  • Referans5 Canal Neto, A., Jorge, F. E. ve De Castro, M. (2002). Improved generator coordinate Hartree-Fock method applied to generate Gaussian basis sets for the isoelectronic series of the atoms He to Ne. Int. J. Quantum Chem., 88(2), 252-262. DOI:https://doi.org/10.1002/qua.10145
  • Referans6 Coşkun, M. ve Ertürk, M. (2022). Comparative performance of different hyperbolic cosine functions and generalized B functions basis sets for atomic systems. Phys. Scripta, 97(7), 1-11. DOI: 10.1088/1402-4896/ac7588
  • Referans7 Coşkun, M. ve Ertürk, M. (2022). Double hyperbolic cosine basis sets for LCAO calculations. Mol. Phys., 120(17), 1-7. DOI: https://doi.org/10.1080/00268976.2022.2109527
  • Referans8 Çopuroğlu, E. (2017).Evaluation of Self-Friction Three-Center Nuclear Attraction Integrals with Integer and Noninteger Principal Quantum Numbers over Slater Type Orbitals. Journal of Chemistry, 2017, 1-6. DOI: https://doi.org/10.1155/2017/1598951
  • Referans9 Ema, I., Garcia de la Vega, J. M. , Miguel, B., Dotterweich, J., Meißner, H. ve Steinborn, E. O. (1999). Expotential-Type Basis Functions: Single-and Double-Zeta B Function Basis Sets for the Ground States of Neutral Atoms from Z = 2 to Z = 36. At. Data Nucl. Data Tables, 72(1), 57-99. DOI: https://doi.org/10.1006/adnd.1999.0809
  • Referans10 Ertürk, M. ve Öztürk, E. (2018). Modified B function basis sets with generalized hyperbolic cosine functions. Comput. Theor. Chem., 1127, 37-43. DOI:https://doi.org/10.1016/j.comptc.2018.02.003
  • Referans11 Ertürk, M. ve Şahin, E. (2020). Generalized B functions applied to atomic calculations. Chem. Phys., 529, 110549. DOI:https://doi.org/10.1016/j.chemphys.2019.110549
  • Referans12 Filter, E. ve Steinborn, E. O. (1978). Extremely compact formulas for molecular two-center one-center integrals and Coulomb integrals over Slater-type atomic orbitals. Phys. Rev. A, 18(1), 1-11. DOI: https://doi.org/10.1103/PhysRevA.18.1
  • Referans13 Guseinov, I. I. (2007). Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems. J. Math. Chem., 42(2), 177-189. DOI: https://doi.org/10.1007/s10910-006-9090-0
  • Referans14 Guseinov, I. I., Mamedov, B. A., Ertürk, M., Aksu, H. ve Sahin, E. (2007). Application of combined Hartree-Fock-Roothaan theory to atoms with an arbitrary number of closed and open shells of any symmetry. Few‐Body Syst., 41(3-4), 211-220. DOI: https://doi.org/10.1007/s00601-007-0179-1
  • Referans15 Kato, T. (1957). On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math., 10(2), 151-177. DOI:https://doi.org/10.1002/cpa.3160100201
  • Referans16 Koga, T., Watanabe, S., Kanayama, K., Yasuda, R. ve Thakkar, A. J. (1995). Improved Roothaan–Hartree–Fock wave functions for atoms and ions with N≤54. J. Chem. Phys., 103(8), 3000-3005. DOI: https://doi.org/10.1063/1.470488
  • Referans17 Koga, T. ve Kanayama, K. (1997). Generalized exponential functions applied to atomic calculations. Z. Phys. D, 41(2), 111-115. DOI: https://doi.org/10.1007/s004600050297
  • Referans18 Mamedov, B. A. ve Çopuroğlu, E. (2011). Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers. Acta Physica Polonica A, 119(3), 332-335. DOI:https://doi.org/10.12693/APhysPolA.119.332
  • Referans19 Mamedov, B. A. ve Çopuroğlu, E. (2012). Calculation of two-center nuclear attraction integrals of Slater type orbitals with noninteger principal quantum numbers using Guseinov’s one-center expansion formulas and Löwdin-α radial function. Applied Mathematics and Computation, 218(15), 7893-7897. DOI: https://doi.org/10.1016/j.amc.2012.01.069
  • Referans20 Moore, C. E. (1970). Ionization potentials and ionization limits derived from the analyses of optical spectra; NSRDSNBS 34; National Bureau of Standards: Washington, DC. Erişim adresi: https://nvlpubs.nist.gov/nistpubs/Legacy/NSRDS/nbsnsrds34.pdf
  • Referans21 Parr, R. G. ve Joy, H. W. (1957). Why Not Use Slater Orbitals of Nonintegral Principal Quantum Number? The Journal of Chemical Physics, 26(2), 424-424. DOI: https://doi.org/10.1063/1.1743314
  • Referans22 Roothaan, C. C. J. (1951). New Developments in Molecular Orbital Theory. Rev. Mod. Phys., 23(2), 69-89. DOI: https://doi.org/10.1103/RevModPhys.23.69
  • Referans23 Roothaan, C. C. J. (1960). Self-Consistent Field Theory for Open Shells of Electronic Systems. Rev. Mod. Phys.,32(2), 179-185. DOI: https://doi.org/10.1103/RevModPhys.32.179
  • Referans24 Slater, j. C. (1930). Atomic Shielding Constants. Phys. Rev., 36(1), 57-64. DOI: https://doi.org/10.1103/PhysRev.36.57
  • Referans25 Snyder, L. C. (1960). Helium Atom Wave Functions from Slater Orbitals of Nonintegral Principal Quantum Number. The Journal ofChem. Phys., 33(6), 1711-1712. DOI: https://doi.org/10.1063/1.1731489
  • Referans26 Steinborn, E. O., Homeier, H. H. H., Fernandez Rico, J., Ema, I., Lopez, R., Ramirez,G. (1999). An improved program for molecular calculations with B functions. J. Mol. Struct. Theochem., 490(1-3), 201-217. DOI: https://doi.org/10.1016/S0166-1280(99)00099-8
  • Referans27 Steinborn, E. O., Homeier, H. H. H., Ema, I., Lopez, R. ve Ramirez, G. (2000). Molecular Calculations with B functions. International Journal of Quantum Chemistery, 76(2), 244-251. DOI: https://doi.org/10.1002/(SICI)1097-461X(2000)76:2<244::AID-QUA13>3.0.CO;2-T
  • Referans28 Tricomi, F. (1947). Sulle funzioni ipergeometriche confluenti. Annali di Matematica Pura ed Applicata, 26(1), 141-175. DOI: http://doi.org/10.1007/BF02415375
  • Referans29 Weniger, E. J. ve Steinborn, E. O. (1983). Numerical properties of the convolution theorems of B functions. Phys. Rev.A, 28(4), 2026-2041. DOI: https://doi.org/10.1103/PhysRevA.28.2026
  • Referans30 Weniger, E. J. (2021). Chapter Ten - Are B functions with nonintegral orders a computationally useful basis set? Adv. Quantum Chem., 83, 209-237. DOI:https://doi.org/10.1016/bs.aiq.2021.06.002

Noninteger Bessel Type Orbitals for Two Electron Atomic Systems

Year 2023, Volume: 9 Issue: 2, 375 - 384, 30.06.2023

Meral Coşkun , Murat Ertürk

https://doi.org/10.28979/jarnas.1163388

Abstract

The aim of this study is to examine the applicability of Bessel type orbitals with non-integer principal quantum numbers to the atomic systems with the Hartree-Fock-Roothaan method and their superiority over other exponential orbitals used in the literature. By the use of combined Hartree-Fock-Roothaan method, the orbital and ground state energy values of two-electron atomic systems have been calculated using the newly proposed Bessel type orbitals. The results obtained with the new basis functions within the minimal basis sets framework are given comparatively in the tables. The results obtained give better values than the similar exponential type basis functions used in the literature and are in good agreement with numerical Hartree-Fock values.

Keywords

Basis functions bessel functions hartree-fock-roothaan method isoelectronic series nonintegral principal quantum number

Project Number

FYL-2019-2851

References

  • Referans1 Agmon, S. (1982). Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations: Bound on Eigen functions of N-Body Schrödinger Operators. Princeton University Press, Princeton. Erişim adresi: https://press.princeton.edu/books/hardcover/9780691641423/lectures-on-exponential-decay-of-solutions-of-second-order-elliptic
  • Referans2 Allouche, A. (1974). Les orbitales de Slater à nombre quantique ≪n≫ non-entier. Theor. Chim. Acta34(1), 79-83. DOI: https://doi.org/10.1007/BF00553235
  • Referans3 Bishop, D. M. ve Leclerc, J. C. (1972). Unconventional basis sets in quantum mechanical calculations. Mol. Phys., 24(5), 979-992. DOI: https://doi.org/10.1080/00268977200102091
  • Referans4 Boys, S. F. (1950). Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system. Proc. Roy. Soc. London A., 200(1063), 542-554. DOI: https://doi.org/10.1098/rspa.1950.0036
  • Referans5 Canal Neto, A., Jorge, F. E. ve De Castro, M. (2002). Improved generator coordinate Hartree-Fock method applied to generate Gaussian basis sets for the isoelectronic series of the atoms He to Ne. Int. J. Quantum Chem., 88(2), 252-262. DOI:https://doi.org/10.1002/qua.10145
  • Referans6 Coşkun, M. ve Ertürk, M. (2022). Comparative performance of different hyperbolic cosine functions and generalized B functions basis sets for atomic systems. Phys. Scripta, 97(7), 1-11. DOI: 10.1088/1402-4896/ac7588
  • Referans7 Coşkun, M. ve Ertürk, M. (2022). Double hyperbolic cosine basis sets for LCAO calculations. Mol. Phys., 120(17), 1-7. DOI: https://doi.org/10.1080/00268976.2022.2109527
  • Referans8 Çopuroğlu, E. (2017).Evaluation of Self-Friction Three-Center Nuclear Attraction Integrals with Integer and Noninteger Principal Quantum Numbers over Slater Type Orbitals. Journal of Chemistry, 2017, 1-6. DOI: https://doi.org/10.1155/2017/1598951
  • Referans9 Ema, I., Garcia de la Vega, J. M. , Miguel, B., Dotterweich, J., Meißner, H. ve Steinborn, E. O. (1999). Expotential-Type Basis Functions: Single-and Double-Zeta B Function Basis Sets for the Ground States of Neutral Atoms from Z = 2 to Z = 36. At. Data Nucl. Data Tables, 72(1), 57-99. DOI: https://doi.org/10.1006/adnd.1999.0809
  • Referans10 Ertürk, M. ve Öztürk, E. (2018). Modified B function basis sets with generalized hyperbolic cosine functions. Comput. Theor. Chem., 1127, 37-43. DOI:https://doi.org/10.1016/j.comptc.2018.02.003
  • Referans11 Ertürk, M. ve Şahin, E. (2020). Generalized B functions applied to atomic calculations. Chem. Phys., 529, 110549. DOI:https://doi.org/10.1016/j.chemphys.2019.110549
  • Referans12 Filter, E. ve Steinborn, E. O. (1978). Extremely compact formulas for molecular two-center one-center integrals and Coulomb integrals over Slater-type atomic orbitals. Phys. Rev. A, 18(1), 1-11. DOI: https://doi.org/10.1103/PhysRevA.18.1
  • Referans13 Guseinov, I. I. (2007). Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems. J. Math. Chem., 42(2), 177-189. DOI: https://doi.org/10.1007/s10910-006-9090-0
  • Referans14 Guseinov, I. I., Mamedov, B. A., Ertürk, M., Aksu, H. ve Sahin, E. (2007). Application of combined Hartree-Fock-Roothaan theory to atoms with an arbitrary number of closed and open shells of any symmetry. Few‐Body Syst., 41(3-4), 211-220. DOI: https://doi.org/10.1007/s00601-007-0179-1
  • Referans15 Kato, T. (1957). On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math., 10(2), 151-177. DOI:https://doi.org/10.1002/cpa.3160100201
  • Referans16 Koga, T., Watanabe, S., Kanayama, K., Yasuda, R. ve Thakkar, A. J. (1995). Improved Roothaan–Hartree–Fock wave functions for atoms and ions with N≤54. J. Chem. Phys., 103(8), 3000-3005. DOI: https://doi.org/10.1063/1.470488
  • Referans17 Koga, T. ve Kanayama, K. (1997). Generalized exponential functions applied to atomic calculations. Z. Phys. D, 41(2), 111-115. DOI: https://doi.org/10.1007/s004600050297
  • Referans18 Mamedov, B. A. ve Çopuroğlu, E. (2011). Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers. Acta Physica Polonica A, 119(3), 332-335. DOI:https://doi.org/10.12693/APhysPolA.119.332
  • Referans19 Mamedov, B. A. ve Çopuroğlu, E. (2012). Calculation of two-center nuclear attraction integrals of Slater type orbitals with noninteger principal quantum numbers using Guseinov’s one-center expansion formulas and Löwdin-α radial function. Applied Mathematics and Computation, 218(15), 7893-7897. DOI: https://doi.org/10.1016/j.amc.2012.01.069
  • Referans20 Moore, C. E. (1970). Ionization potentials and ionization limits derived from the analyses of optical spectra; NSRDSNBS 34; National Bureau of Standards: Washington, DC. Erişim adresi: https://nvlpubs.nist.gov/nistpubs/Legacy/NSRDS/nbsnsrds34.pdf
  • Referans21 Parr, R. G. ve Joy, H. W. (1957). Why Not Use Slater Orbitals of Nonintegral Principal Quantum Number? The Journal of Chemical Physics, 26(2), 424-424. DOI: https://doi.org/10.1063/1.1743314
  • Referans22 Roothaan, C. C. J. (1951). New Developments in Molecular Orbital Theory. Rev. Mod. Phys., 23(2), 69-89. DOI: https://doi.org/10.1103/RevModPhys.23.69
  • Referans23 Roothaan, C. C. J. (1960). Self-Consistent Field Theory for Open Shells of Electronic Systems. Rev. Mod. Phys.,32(2), 179-185. DOI: https://doi.org/10.1103/RevModPhys.32.179
  • Referans24 Slater, j. C. (1930). Atomic Shielding Constants. Phys. Rev., 36(1), 57-64. DOI: https://doi.org/10.1103/PhysRev.36.57
  • Referans25 Snyder, L. C. (1960). Helium Atom Wave Functions from Slater Orbitals of Nonintegral Principal Quantum Number. The Journal ofChem. Phys., 33(6), 1711-1712. DOI: https://doi.org/10.1063/1.1731489
  • Referans26 Steinborn, E. O., Homeier, H. H. H., Fernandez Rico, J., Ema, I., Lopez, R., Ramirez,G. (1999). An improved program for molecular calculations with B functions. J. Mol. Struct. Theochem., 490(1-3), 201-217. DOI: https://doi.org/10.1016/S0166-1280(99)00099-8
  • Referans27 Steinborn, E. O., Homeier, H. H. H., Ema, I., Lopez, R. ve Ramirez, G. (2000). Molecular Calculations with B functions. International Journal of Quantum Chemistery, 76(2), 244-251. DOI: https://doi.org/10.1002/(SICI)1097-461X(2000)76:2<244::AID-QUA13>3.0.CO;2-T
  • Referans28 Tricomi, F. (1947). Sulle funzioni ipergeometriche confluenti. Annali di Matematica Pura ed Applicata, 26(1), 141-175. DOI: http://doi.org/10.1007/BF02415375
  • Referans29 Weniger, E. J. ve Steinborn, E. O. (1983). Numerical properties of the convolution theorems of B functions. Phys. Rev.A, 28(4), 2026-2041. DOI: https://doi.org/10.1103/PhysRevA.28.2026
  • Referans30 Weniger, E. J. (2021). Chapter Ten - Are B functions with nonintegral orders a computationally useful basis set? Adv. Quantum Chem., 83, 209-237. DOI:https://doi.org/10.1016/bs.aiq.2021.06.002
There are 30 citations in total.

Details

Primary Language Turkish
Subjects Atomic, Molecular and Optical Physics
Journal Section Research Article
Authors

Meral Coşkun 0000-0003-1167-1697

Murat Ertürk 0000-0002-7039-1970

Project Number FYL-2019-2851
Early Pub Date June 21, 2023
Publication Date June 30, 2023
Submission Date August 17, 2022
Published in Issue Year 2023 Volume: 9 Issue: 2

Cite

APA Coşkun, M., & Ertürk, M. (2023). İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller. Journal of Advanced Research in Natural and Applied Sciences, 9(2), 375-384. https://doi.org/10.28979/jarnas.1163388
AMA Coşkun M, Ertürk M. İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller. JARNAS. June 2023;9(2):375-384. doi:10.28979/jarnas.1163388
Chicago Coşkun, Meral, and Murat Ertürk. “İki Elektronlu Atomik Sistemler için Baş Kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller”. Journal of Advanced Research in Natural and Applied Sciences 9, no. 2 (June 2023): 375-84. https://doi.org/10.28979/jarnas.1163388.
EndNote Coşkun M, Ertürk M (June 1, 2023) İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller. Journal of Advanced Research in Natural and Applied Sciences 9 2 375–384.
IEEE M. Coşkun and M. Ertürk, “İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller”, JARNAS, vol. 9, no. 2, pp. 375–384, 2023, doi: 10.28979/jarnas.1163388.
ISNAD Coşkun, Meral - Ertürk, Murat. “İki Elektronlu Atomik Sistemler için Baş Kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller”. Journal of Advanced Research in Natural and Applied Sciences 9/2 (June 2023), 375-384. https://doi.org/10.28979/jarnas.1163388.
JAMA Coşkun M, Ertürk M. İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller. JARNAS. 2023;9:375–384.
MLA Coşkun, Meral and Murat Ertürk. “İki Elektronlu Atomik Sistemler için Baş Kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller”. Journal of Advanced Research in Natural and Applied Sciences, vol. 9, no. 2, 2023, pp. 375-84, doi:10.28979/jarnas.1163388.
Vancouver Coşkun M, Ertürk M. İki Elektronlu Atomik Sistemler için Baş kuantum Sayısı Kesir Değerli Bessel Tipli Orbitaller. JARNAS. 2023;9(2):375-84.


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