A Scheme for Calculating Atomic Structures beyond the Spherical Approximation

ABSTRACT

We present a scheme for calculating atomic single-particle wave functions and spectra with taking into ac-count the nonspherical effect explicitly. The actual calculation is also performed for the neutral carbon atom within the Hartree-Fock-Slater approximation. As compared with the conventional atomic structure of the spherical approximation, the degenerate energy levels are split partially. The ground state values of the total orbital and spin angular momenta are estimated to be both about unity, which corresponds to the term P3PP in the LS-multiplet theory. This means that the nonspherical effect may play an essential role on the description of the magnetization caused by the orbital polarization.

We present a scheme for calculating atomic single-particle wave functions and spectra with taking into ac-count the nonspherical effect explicitly. The actual calculation is also performed for the neutral carbon atom within the Hartree-Fock-Slater approximation. As compared with the conventional atomic structure of the spherical approximation, the degenerate energy levels are split partially. The ground state values of the total orbital and spin angular momenta are estimated to be both about unity, which corresponds to the term P3PP in the LS-multiplet theory. This means that the nonspherical effect may play an essential role on the description of the magnetization caused by the orbital polarization.

KEYWORDS

Nonspherical Distribution Of Electrons, Spherical Approximation, Orbital Polarization, Atomic Structure, Carbon Atom

Nonspherical Distribution Of Electrons, Spherical Approximation, Orbital Polarization, Atomic Structure, Carbon Atom

Cite this paper

nullM. Miyasita, K. Higuchi and M. Higuchi, "A Scheme for Calculating Atomic Structures beyond the Spherical Approximation,"*Journal of Modern Physics*, Vol. 2 No. 5, 2011, pp. 421-430. doi: 10.4236/jmp.2011.25052.

nullM. Miyasita, K. Higuchi and M. Higuchi, "A Scheme for Calculating Atomic Structures beyond the Spherical Approximation,"

References

[1] D. R. Hartree, “The Wave Mechanics of an Atom with a Noncoulomb Central Field. PartI: Theory and Method. Part II: Some Results and Discussions,” Proceedings of Cambridge Philosophical Society, Vol. 24, No. 1, 1928, 111-132.

[2] V. Fock, Z. Physik, “N?herungsmethode zur L?sung des quantenmechanischen Mehrk?rperproblems,” Zeitschrift Für Physik, Vol. 61, No. 1-2, pp. 126-148. doi:10.1007/BF01340294

[3] P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Physical Review, Vol. 136, No. 3B, 1964, pp. 864-871. doi:10.1103/PhysRev.136.B864

[4] W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Physical Review, Vol. 140, No. 4A, 1965, pp. 1133-1138. doi:10.1103/PhysRev.140.A1133

[5] The spherical approximation means that the effective potential in Equation (3) is approximated into the central field, i.e., . For instance, see, J. C. Slater, “Quantum Theory of Atomic Structure,” McGraw-Hill, NY, Vol. 1, 1960.

[6] This means that .

[7] A. Messiah, “Quantum Mechanics”, Dover Publications, NY, 1999.

[8] F. Herman and S. Skillman, “Atomic Structure Calculations,” Prentice-Hall Inc., New Jersey, 1963.

[9] J. C. Slater, “The Calculation of Molecular Orbitals,” John Wiley & Sons, NY, 1979.

[10] N. F. Mott, “The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals,” Proceedings of the Physical Society, London, Section A, Vol. 62, No. 7, 1949, p. 416. doi: 10.1088/0370-1298/62/7/303

[11] P. W. Anderson, “New Approach to the Theory of Superexchange Interactions,” Physical Review, Vol. 115, No. 1, 1959, pp. 2-13. doi: 10.1103/PhysRev.115.2

[12] V. I. Anisimov, J. Zaanen and O. K. Andersen, “Band Theory and Mott Insulators: Hubbard U instead of Stoner,” Physical Review B, Vol. 44, No. 3, 1991, pp. 943-954. doi:10.1103/PhysRevB.44.943

[13] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, “Density-Functional Theory and Strong Interations: Orbital Ordering in Mott-Hubbard Insulators,” Physical Review B, Vol. 52, No. 8, 1995, R5468-R5470.

[14] S. E. Koonin, “Computational Physics,” Addison-Wesley, NY, 1986.

[15] O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring and M. S. S. Brooks, “Orbital magnetism in Fe, Co, and Ni,” Physical Review B, Vol. 42, 1990, 2707-2710. doi:10.1103/PhysRevB.42.2707

[16] O. Eriksson, M. S. S. Brooks and B. Johansson, Phys. Rev. B 41, 7311 (1990). doi: 10.1103/PhysRevB.42.2707

[17] M. S. S. Brooks, O. Eriksson, L. Severin and B. Johansson, “Spin and Orbital Magnetization Densities in Itinerant Magnets” Physica B, Vol. 192, No. 1-2, 1993, pp. 39-49. doi:10.1016/0921-4526(93)90106-G

[18] T. Shishidou, T. Oguchi and T. Jo, “Hartree-Fock Study on the 5f Orbital Magnetic Moment of US,” Physical Review B, Vol. 59, No. 10, 1999, pp. 6813-6823. doi:10.1103/PhysRevB.59.6813

[19] M. R. Norman, “Orbital polarization and the insulating gap in the transition-metal oxides,” Physical Review Letters, Vol. 64, No. 10, 1990, pp. 1162-1165. doi:10.1103/PhysRevLett.64.1162

[20] G. H. Daalderop, P. J. Kelly and M. F. H. Schuurmans, “Magnetocrystalline Anisotropy and Orbital Moments in Transition-Metal Compounds,” Physical Review B, Vol. 44, No. 21, 1992, pp. 12054-12057. doi:10.1103/PhysRevB.44.12054

[21] A. Narita and M. Higuchi, “Expressions of Energy and Potential due to Orbital Polarization,” Journal of the Physical Society of Japan, Vol. 75, No. 2, 2006, pp. 024301-024301-10. doi:10.1143/JPSJ.75.024301

[22] J. F. Janak and A. R. Williams, “Method for Calculating Wave Functions in a Nonspherical Potential,” Physical Review B, Vol. 23, No. 12, 1981, pp. 6301-6306. doi:10.1103/PhysRevB.23.6301

[23] F. W. Kutzler and G. S. Painter, “Energies of Atoms with Nonspherical Charge Densities Calculated with Nonlocal Density-Functional Theory,” Physical Review Letters, Vol. 59, No. 12, 1987, pp. 1285-1288. doi:10.1103/PhysRevLett.59.1285

[24] A. D. Becke, “Local Exchange-Correlation Approximations and First-Row Molecular Dissociation Energies,” International Journal of Quantum Chemistry, Vol. 27, No. 5, 1985, pp. 585-594. doi:10.1002/qua.560270507

[25] A. D. Becke, “Current Density in Exchange-Correlation Functionals: Application to Atomic States,” Journal of Chemical Physics, Vol. 117, No. 15, 2002, pp. 6935-6938. doi:10.1063/1.1503772

[26] E. Orestes, T. Marcasso and K. Capelle, “Density-Functional Calculation of Ionization Energies of Current-Carrying Atomic States,” Physical Review A, Vol. 68, No. 2, 2003, 022105. doi:10.1103/PhysRevA.68.022105

[27] E. Orestes, A. B. F. da Silva and K. Capelle, “Energy Lowering of Current-Carrying Single-Particle States in Open-Shell atoms due to an Exchange-Correlation Vector Potential,” International Journal Of Quantum Chemistry, Vol. 103, No. 5, 2005, pp. 516-522. doi:10.1002/qua.20575

[28] G. Vignale and M. Rasolt, “Density-Functional Theory in Strong Magnetic Fields,” Physical Review Letters, Vol. 59, No. 20, 1987, pp. 2360-2363. doi:10.1103/PhysRevLett.59.2360

[29] G. Vignale and M. Rasolt, “Current- and Spin-Density- Functional Theory for Inhomogeneous Electronic Systems in Strong Magnetic Fields,” Physical Review B, Vol. 37, No. 18, 1988, 10685-10696. doi:10.1103/PhysRevB.37.10685

[30] M. Higuchi and A. Hasegawa, “A Relativistic Current- and Spin-Density Functional Theory and a Single-Particle Equation,” Journal of the Physical Society of Japan, Vol. 66, No. 1, 1997, p. 149 (1997). doi:10.1143/JPSJ.66.149

[31] M. Higuchi and A. Hasegawa, “Single-Particle Equation of Relativistic Current- and Spin-Density Functional Theory and Its Application to the Atomic Structure of the Lanthanide Series,” Journal of the Physical Society of Japan, Vol. 67, No. 6, 1998, pp. 2037-2047. doi:10.1143/JPSJ.67.2037

[32] J. C. Slater, “A Simplification of the Hartree-Fock Method,” Physical Review, Vol. 81, No. 3, 1951, pp. 385-390.doi: 10.1103/PhysRev.81.385

[1] D. R. Hartree, “The Wave Mechanics of an Atom with a Noncoulomb Central Field. PartI: Theory and Method. Part II: Some Results and Discussions,” Proceedings of Cambridge Philosophical Society, Vol. 24, No. 1, 1928, 111-132.

[2] V. Fock, Z. Physik, “N?herungsmethode zur L?sung des quantenmechanischen Mehrk?rperproblems,” Zeitschrift Für Physik, Vol. 61, No. 1-2, pp. 126-148. doi:10.1007/BF01340294

[3] P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Physical Review, Vol. 136, No. 3B, 1964, pp. 864-871. doi:10.1103/PhysRev.136.B864

[4] W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Physical Review, Vol. 140, No. 4A, 1965, pp. 1133-1138. doi:10.1103/PhysRev.140.A1133

[5] The spherical approximation means that the effective potential in Equation (3) is approximated into the central field, i.e., . For instance, see, J. C. Slater, “Quantum Theory of Atomic Structure,” McGraw-Hill, NY, Vol. 1, 1960.

[6] This means that .

[7] A. Messiah, “Quantum Mechanics”, Dover Publications, NY, 1999.

[8] F. Herman and S. Skillman, “Atomic Structure Calculations,” Prentice-Hall Inc., New Jersey, 1963.

[9] J. C. Slater, “The Calculation of Molecular Orbitals,” John Wiley & Sons, NY, 1979.

[10] N. F. Mott, “The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals,” Proceedings of the Physical Society, London, Section A, Vol. 62, No. 7, 1949, p. 416. doi: 10.1088/0370-1298/62/7/303

[11] P. W. Anderson, “New Approach to the Theory of Superexchange Interactions,” Physical Review, Vol. 115, No. 1, 1959, pp. 2-13. doi: 10.1103/PhysRev.115.2

[12] V. I. Anisimov, J. Zaanen and O. K. Andersen, “Band Theory and Mott Insulators: Hubbard U instead of Stoner,” Physical Review B, Vol. 44, No. 3, 1991, pp. 943-954. doi:10.1103/PhysRevB.44.943

[13] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, “Density-Functional Theory and Strong Interations: Orbital Ordering in Mott-Hubbard Insulators,” Physical Review B, Vol. 52, No. 8, 1995, R5468-R5470.

[14] S. E. Koonin, “Computational Physics,” Addison-Wesley, NY, 1986.

[15] O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring and M. S. S. Brooks, “Orbital magnetism in Fe, Co, and Ni,” Physical Review B, Vol. 42, 1990, 2707-2710. doi:10.1103/PhysRevB.42.2707

[16] O. Eriksson, M. S. S. Brooks and B. Johansson, Phys. Rev. B 41, 7311 (1990). doi: 10.1103/PhysRevB.42.2707

[17] M. S. S. Brooks, O. Eriksson, L. Severin and B. Johansson, “Spin and Orbital Magnetization Densities in Itinerant Magnets” Physica B, Vol. 192, No. 1-2, 1993, pp. 39-49. doi:10.1016/0921-4526(93)90106-G

[18] T. Shishidou, T. Oguchi and T. Jo, “Hartree-Fock Study on the 5f Orbital Magnetic Moment of US,” Physical Review B, Vol. 59, No. 10, 1999, pp. 6813-6823. doi:10.1103/PhysRevB.59.6813

[19] M. R. Norman, “Orbital polarization and the insulating gap in the transition-metal oxides,” Physical Review Letters, Vol. 64, No. 10, 1990, pp. 1162-1165. doi:10.1103/PhysRevLett.64.1162

[20] G. H. Daalderop, P. J. Kelly and M. F. H. Schuurmans, “Magnetocrystalline Anisotropy and Orbital Moments in Transition-Metal Compounds,” Physical Review B, Vol. 44, No. 21, 1992, pp. 12054-12057. doi:10.1103/PhysRevB.44.12054

[21] A. Narita and M. Higuchi, “Expressions of Energy and Potential due to Orbital Polarization,” Journal of the Physical Society of Japan, Vol. 75, No. 2, 2006, pp. 024301-024301-10. doi:10.1143/JPSJ.75.024301

[22] J. F. Janak and A. R. Williams, “Method for Calculating Wave Functions in a Nonspherical Potential,” Physical Review B, Vol. 23, No. 12, 1981, pp. 6301-6306. doi:10.1103/PhysRevB.23.6301

[23] F. W. Kutzler and G. S. Painter, “Energies of Atoms with Nonspherical Charge Densities Calculated with Nonlocal Density-Functional Theory,” Physical Review Letters, Vol. 59, No. 12, 1987, pp. 1285-1288. doi:10.1103/PhysRevLett.59.1285

[24] A. D. Becke, “Local Exchange-Correlation Approximations and First-Row Molecular Dissociation Energies,” International Journal of Quantum Chemistry, Vol. 27, No. 5, 1985, pp. 585-594. doi:10.1002/qua.560270507

[25] A. D. Becke, “Current Density in Exchange-Correlation Functionals: Application to Atomic States,” Journal of Chemical Physics, Vol. 117, No. 15, 2002, pp. 6935-6938. doi:10.1063/1.1503772

[26] E. Orestes, T. Marcasso and K. Capelle, “Density-Functional Calculation of Ionization Energies of Current-Carrying Atomic States,” Physical Review A, Vol. 68, No. 2, 2003, 022105. doi:10.1103/PhysRevA.68.022105

[27] E. Orestes, A. B. F. da Silva and K. Capelle, “Energy Lowering of Current-Carrying Single-Particle States in Open-Shell atoms due to an Exchange-Correlation Vector Potential,” International Journal Of Quantum Chemistry, Vol. 103, No. 5, 2005, pp. 516-522. doi:10.1002/qua.20575

[28] G. Vignale and M. Rasolt, “Density-Functional Theory in Strong Magnetic Fields,” Physical Review Letters, Vol. 59, No. 20, 1987, pp. 2360-2363. doi:10.1103/PhysRevLett.59.2360

[29] G. Vignale and M. Rasolt, “Current- and Spin-Density- Functional Theory for Inhomogeneous Electronic Systems in Strong Magnetic Fields,” Physical Review B, Vol. 37, No. 18, 1988, 10685-10696. doi:10.1103/PhysRevB.37.10685

[30] M. Higuchi and A. Hasegawa, “A Relativistic Current- and Spin-Density Functional Theory and a Single-Particle Equation,” Journal of the Physical Society of Japan, Vol. 66, No. 1, 1997, p. 149 (1997). doi:10.1143/JPSJ.66.149

[31] M. Higuchi and A. Hasegawa, “Single-Particle Equation of Relativistic Current- and Spin-Density Functional Theory and Its Application to the Atomic Structure of the Lanthanide Series,” Journal of the Physical Society of Japan, Vol. 67, No. 6, 1998, pp. 2037-2047. doi:10.1143/JPSJ.67.2037

[32] J. C. Slater, “A Simplification of the Hartree-Fock Method,” Physical Review, Vol. 81, No. 3, 1951, pp. 385-390.doi: 10.1103/PhysRev.81.385