Gaussian-Transform for the Dirac Wave Function and its Application to the Multicenter Molecular Integral Over Dirac Wave Functions for Solving the Molecular Matrix Dirac Equation

Kazuhiro Ishida

doi:10.61927/igmin266

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Biology Group Research Article 記事ID: igmin266

Gaussian-Transform for the Dirac Wave Function and its Application to the Multicenter Molecular Integral Over Dirac Wave Functions for Solving the Molecular Matrix Dirac Equation

Biostatistics Biophysics

Kazuhiro Ishida ^*

Affiliation

219-48 Matsugasaki, Kashiwa City, Chiba 277-0835, Japan

DOI10.61927/igmin266 全文HTML 全文PDF この記事を引用する

要約

Gaussian-transform formula is derived for the Dirac wave function. Using it, one can derive the multicenter molecular integral over Dirac wave functions for any physical quantity. As the first application of it, multicenter molecular integrals over Dirac wave functions are derived for the homogeneous charge density distribution model and the Gauss-type charge density distribution model. Such integrals are necessary for solving the gauge-invariant molecular matrix Dirac equation with using the restricted magnetic balance.

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参考文献

WM, Sun XS, Chen XF, Liu, Wang F. Gauge-invariant hydrogen-atom Hamiltonian. Phys Rev A. 2010;82:012107.
Komorovský S, Repiský M, Malkina OL, Malkin VG. Fully relativistic calculations of NMR shielding tensors using restricted magnetically balanced basis and gauge including atomic orbitals. J Chem Phys. 2010 Apr 21;132(15):154101. doi: 10.1063/1.3359849. PMID: 20423162.
Yoshizawa T. On the development of the exact two-component relativistic method for calculating indirect NMR spin-spin coupling constants. Chem Phys. 2019;518:112-122.
Fukui H, Baba T, Shiraishi Y, Imanishi S, Kubo K, Mari K, Shimoji M. Calculation of nuclear magnetic shieldings: infinite-order Foldy-Wouthuysen transformation. Mol Phys. 2004;102:641-648.
Andrae D. Nuclear charge density distribution in quantum chemistry. In: Schwerdtfeger P, editor. Relativistic Electronic Structure Theory Part 1. Amsterdam: Elsevier; 2002;203-258.
Visscher L, Dyall KG. Dirac-Fock atomic structure calculations using different nuclear charge distributions. At Data Nucl Data Tables. 1997;67:207-224.
Hennum AC, Klopper W, Helgaker T. Direct perturbation theory of magnetic properties and relativistic corrections for the point nuclear and Gaussian nuclear models. J Chem Phys. 2001;115:7356-7363.
Kobus J, Quiney HM, Wilson S. A comparison of finite difference and finite basis set Hartree-Fock calculations for the N2 molecule with finite nuclei. J Phys B. 2001;34:2045-2056.
Ishida K. A reason why to use the Gaussian-type-orbital is not suitable for the relativistic calculation of the nuclear-magnetic-resonance spectra with using the restricted magnetic balance, Comput Theor Chem 2024;1241:114804
Shavitt I, Karplus M. Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J Chem Phys. 1965;43:398-414.
Ishida K. Calculus of several harmonic functions. J Comput Chem Jpn, Int Ed. 2022;8:2021-0029.
Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series, and Products. New York: Academic Press; 2007. Formula # 3.471.3.
Silverstone HJ. On the evaluation of two-center overlap and Coulomb integral with non-integer n Slater-type orbitals. J Chem Phys. 1966;45:4337-4341.
Petersson GA, McKoy V. Application of non-integer transformation of multicenter integrals. J Chem Phys. 1967;46:4362-4368.
Guseinov II, Mamedov BA. Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals. J Chem Phys. 2004 Jul 22;121(4):1649-54. doi: 10.1063/1.1766011. PMID: 15260714.
Ozdogan T. Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater-type orbitals with noninteger principal quantum numbers. Int J Quantum Chem. 2003;92:419-427.
Sack RA. Generalization of Laplace’s expansion to arbitrary powers and functions of the distance between two points. J Math Phys. 1964;5:245-251. doi:10.1063/1.1704114.
Abramowitz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover Publications, Inc.; 1972.
Ishida K. Rigorous and rapid calculation of the electron repulsion integral over the uncontracted solid harmonic Gaussian-type orbitals. J Chem Phys. 1999;111:4913-4922.
Ishida K. General formula evaluation of the electron-repulsion integrals and the first and second derivatives over Gaussian-type orbitals. J Chem Phys. 1991;95:5198-5205.

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[1] WM, Sun XS, Chen XF, Liu, Wang F. Gauge-invariant hydrogen-atom Hamiltonian. Phys Rev A. 2010;82:012107.

[2] Komorovský S, Repiský M, Malkina OL, Malkin VG. Fully relativistic calculations of NMR shielding tensors using restricted magnetically balanced basis and gauge including atomic orbitals. J Chem Phys. 2010 Apr 21;132(15):154101. doi: 10.1063/1.3359849. PMID: 20423162.

[3] Yoshizawa T. On the development of the exact two-component relativistic method for calculating indirect NMR spin-spin coupling constants. Chem Phys. 2019;518:112-122.

[4] Fukui H, Baba T, Shiraishi Y, Imanishi S, Kubo K, Mari K, Shimoji M. Calculation of nuclear magnetic shieldings: infinite-order Foldy-Wouthuysen transformation. Mol Phys. 2004;102:641-648.

[5] Andrae D. Nuclear charge density distribution in quantum chemistry. In: Schwerdtfeger P, editor. Relativistic Electronic Structure Theory Part 1. Amsterdam: Elsevier; 2002;203-258.

[6] Visscher L, Dyall KG. Dirac-Fock atomic structure calculations using different nuclear charge distributions. At Data Nucl Data Tables. 1997;67:207-224.

[7] Hennum AC, Klopper W, Helgaker T. Direct perturbation theory of magnetic properties and relativistic corrections for the point nuclear and Gaussian nuclear models. J Chem Phys. 2001;115:7356-7363.

[8] Kobus J, Quiney HM, Wilson S. A comparison of finite difference and finite basis set Hartree-Fock calculations for the N2 molecule with finite nuclei. J Phys B. 2001;34:2045-2056.

[9] Ishida K. A reason why to use the Gaussian-type-orbital is not suitable for the relativistic calculation of the nuclear-magnetic-resonance spectra with using the restricted magnetic balance, Comput Theor Chem 2024;1241:114804

[10] Shavitt I, Karplus M. Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J Chem Phys. 1965;43:398-414.

[11] Ishida K. Calculus of several harmonic functions. J Comput Chem Jpn, Int Ed. 2022;8:2021-0029.

[12] Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series, and Products. New York: Academic Press; 2007. Formula # 3.471.3.

[13] Silverstone HJ. On the evaluation of two-center overlap and Coulomb integral with non-integer n Slater-type orbitals. J Chem Phys. 1966;45:4337-4341.

[14] Petersson GA, McKoy V. Application of non-integer transformation of multicenter integrals. J Chem Phys. 1967;46:4362-4368.

[15] Guseinov II, Mamedov BA. Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals. J Chem Phys. 2004 Jul 22;121(4):1649-54. doi: 10.1063/1.1766011. PMID: 15260714.

[16] Ozdogan T. Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater-type orbitals with noninteger principal quantum numbers. Int J Quantum Chem. 2003;92:419-427.

[17] Sack RA. Generalization of Laplace’s expansion to arbitrary powers and functions of the distance between two points. J Math Phys. 1964;5:245-251. doi:10.1063/1.1704114.

[18] Abramowitz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover Publications, Inc.; 1972.

[19] Ishida K. Rigorous and rapid calculation of the electron repulsion integral over the uncontracted solid harmonic Gaussian-type orbitals. J Chem Phys. 1999;111:4913-4922.

[20] Ishida K. General formula evaluation of the electron-repulsion integrals and the first and second derivatives over Gaussian-type orbitals. J Chem Phys. 1991;95:5198-5205.

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Gaussian-Transform for the Dirac Wave Function and its Application to the Multicenter Molecular Integral Over Dirac Wave Functions for Solving the Molecular Matrix Dirac Equation

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