najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2019-10-1-31-41najo-530Research ArticlePHYSICSФИЗИКАHund’s rule in open-shell states of two-electron systems: From free through confined and screened atoms, to quantum dotsKatrielJacob

Haifa, 32000

jkatriel@technion.ac.ilMontgomeryH. E.

Danville, Kentucky 40422

ed.montgomery@centre.eduSarsaJr., A.

E-14071 C ´ordoba 

fa1sarua@uco.esBuend´ıaE.

E-18071 Granada 

buendia@ugr.esGalvezF. J.

E-18071 Granada 

galvez@ugr.esSenK. D.

Hyderabad-500 046

kds77@uohyd.ac.inDepartment of Chemistry, Technion - Israel Institute of TechnologyIsraelChemistry Program, Centre CollegeUnited StatesDepartamento de F´ısica, Campus de Rabanales Edif. C2, Universidad de CordobaSpainDepartamento de F´ısica Atomica, Molecular y Nuclear, Facultad de Ciencias, ´Universidad de GranadaSpainSchool of Chemistry, University of HyderabadIndia2019060820251013141Copyright © Katriel J., Montgomery H.E., Sarsa J.A., Buend´ıa E., Galvez F., Sen K.D., 20252025Katriel J., Montgomery H.E., Sarsa J.A., Buend´ıa E., Galvez F., Sen K.D.Katriel J., Montgomery H.E., Sarsa J.A., Buend´ıa E., Galvez F., Sen K.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/530

Singly-excited singlet-triplet pairs of states of two-electron spherically symmetric systems, that are degenerate in the absence of inter-electronic repulsion, are revisited. In addition to the obvious two-electron atom we consider the two-electron quantum dot confined by either a harmonic potential or by an infinite spherical well, the confined two-electron atom, and three variants of an atom immersed in a plasma, modeled by the screened Coulomb (Debye) potential. The validity of Hund’s multiplicity rule is confirmed, and the contribution of the interparticle repulsion energy to the singlet-triplet splitting is examined. One feature that all these systems share is that the triplet wave function is contracted relative to that of the corresponding singlet. This feature, which is a consequence of the virial theorem, affects both the behavior of the outer electron ionization energies and the relative magnitudes of the inter-particle repulsion energies in the singlet vs. the triplet. Whereas in atomic highly positive ions the interelectronic repulsion is lower in the triplet than in the corresponding singlet state, this ordering is reversed in neutral atoms. Such reversal does not take place in quantum dots. Confined and screened systems exhibit more nuanced behavior. The analysis utilizes appropriate variants of the virial and Hellmann–Feynman theorems.

two-electron quantum dotsconfined two-electron atomscreened coulomb potentialHund’s multiplicity ruleVirial theoremHellmann-Feynman theoremReferencesHund F., Zur deutung verwickelter spektren, insbesondere der elemente Scandium bis Nickel, Z. Phys., 1925, 33, P. 345–371.Hund F., Zur deutung verwickelter spektren, insbesondere der elemente Scandium bis Nickel, Z. Phys., 1925, 33, P. 345–371.Slater J.C. The theory of complex spectra. Phys. Rev., 1929, 34, P. 1293–1322.Slater J.C. The theory of complex spectra. Phys. Rev., 1929, 34, P. 1293–1322.Morgan III J.D., Kutzelnigg W. Hund’s rules, the alternating rule, and symmetry holes. J. Phys. Chem., 1993, 97, P. 2425–2434.Morgan III J.D., Kutzelnigg W. Hund’s rules, the alternating rule, and symmetry holes. J. Phys. Chem., 1993, 97, P. 2425–2434.Kutzelnigg W., Morgan III J.D. Hund’s rules. Z. Phys. D, 1996, 36, P. 197–214.Kutzelnigg W., Morgan III J.D. Hund’s rules. Z. Phys. D, 1996, 36, P. 197–214.Davidson E.R. Single-configuration calculations on excited states of Helium. II. J. Chem. Phys., 1965, 42, P. 4199–4200.Davidson E.R. Single-configuration calculations on excited states of Helium. II. J. Chem. Phys., 1965, 42, P. 4199–4200.Koga T., Matsuyama H., J. Molina Molina, Dehesa J.S. Electron-pair densities of group 2 atoms in their 1P and 3P terms. Eur. Phys. J. D, 1999, 7, P. 17–23.Koga T., Matsuyama H., J. Molina Molina, Dehesa J.S. Electron-pair densities of group 2 atoms in their 1P and 3P terms. Eur. Phys. J. D, 1999, 7, P. 17–23.Katriel J. A study of the interpretation of Hund’s rule. Theoret. Chim. Acta, 1972, 23, P. 309–315. An interpretation of Hund’s rule, ibid., 1972, 26, P. 163–170.Katriel J. A study of the interpretation of Hund’s rule. Theoret. Chim. Acta, 1972, 23, P. 309–315. An interpretation of Hund’s rule, ibid., 1972, 26, P. 163–170.Katriel J. and Pauncz R., Theoretical interpratation of Hund’s rule. Adv. Quantum Chem., 1977, 10, P. 143–185.Katriel J. and Pauncz R., Theoretical interpratation of Hund’s rule. Adv. Quantum Chem., 1977, 10, P. 143–185.Katriel J. and Themelis S.I. Hund’s rule in the doubly excited states of the Helium isoelectronic sequence. Int. J. Quantum Chem., 2012, 112, P. 2880–2893.Katriel J. and Themelis S.I. Hund’s rule in the doubly excited states of the Helium isoelectronic sequence. Int. J. Quantum Chem., 2012, 112, P. 2880–2893.Koga T. and Koshida Y. Roles of nuclear attraction and electron repulsion energies in the relative stability of atomic LS terms. J. Chem. Phys., 1999, 111, P. 53–54.Koga T. and Koshida Y. Roles of nuclear attraction and electron repulsion energies in the relative stability of atomic LS terms. J. Chem. Phys., 1999, 111, P. 53–54.Katriel J. and Montgomery Jr.H.E. Atomic vs. quantum dot open shell spectr. J. Chem. Phys., 2017, 146, P. 064104-(1-10).Katriel J. and Montgomery Jr.H.E. Atomic vs. quantum dot open shell spectr. J. Chem. Phys., 2017, 146, P. 064104-(1-10).Sako T., Paldus J., Ichimura A. and Diercksen G.H.F. Origin of the first Hund rule and the structure of Fermi holes in two-dimensional He-like atoms and two-electron quantum dots. J. Phys. B, 2012, 45, P. 235001(1–13).Sako T., Paldus J., Ichimura A. and Diercksen G.H.F. Origin of the first Hund rule and the structure of Fermi holes in two-dimensional He-like atoms and two-electron quantum dots. J. Phys. B, 2012, 45, P. 235001(1–13).Katriel J. and Montgomery Jr.H.E., Hund’s rule in the two-electron quantum dot. Eur. Phys. J. B., 2012, 85, P. 394(1–4).Katriel J. and Montgomery Jr.H.E., Hund’s rule in the two-electron quantum dot. Eur. Phys. J. B., 2012, 85, P. 394(1–4).Sarsa A., Buend´ıa E., Galvez F.J. and Katriel J. Singlet ´ vs. triplet interelectronic repulsion in confined atoms. Chem. Phys. Letters, 2018, 702, P. 106–110.Sarsa A., Buend´ıa E., Galvez F.J. and Katriel J. Singlet ´ vs. triplet interelectronic repulsion in confined atoms. Chem. Phys. Letters, 2018, 702, P. 106–110.Katriel J., Montgomery Jr.H.E. and Sen K.D. Hund’s rule in the (1s2s)1;3S states of the two-electron Debye atom. Phys. Plasmas, 2018, 25, P. 092111(1–6).Katriel J., Montgomery Jr.H.E. and Sen K.D. Hund’s rule in the (1s2s)1;3S states of the two-electron Debye atom. Phys. Plasmas, 2018, 25, P. 092111(1–6).Pupyshev V.I. and Montgomery Jr.H.E. Spherically symmetric states of Hookium in a cavity, Phys. Scr., 2015, 90, P. 085401(1–8).Pupyshev V.I. and Montgomery Jr.H.E. Spherically symmetric states of Hookium in a cavity, Phys. Scr., 2015, 90, P. 085401(1–8).Ugalde J.M., Sarasola C. and Lopez X. Atomic and molecular bound ground states of the Yukawa potential. Phys. Rev. A, 1997, 56, P. 1642–1645.Ugalde J.M., Sarasola C. and Lopez X. Atomic and molecular bound ground states of the Yukawa potential. Phys. Rev. A, 1997, 56, P. 1642–1645.Mercero J.M., Fowler J.E., Sarasola C. and Ugalde J.M. Bound excited states of H− and He− in the statically screened Coulomb potential. Phys. Rev. A, 1998, 57, P. 2550–2555.Mercero J.M., Fowler J.E., Sarasola C. and Ugalde J.M. Bound excited states of H− and He− in the statically screened Coulomb potential. Phys. Rev. A, 1998, 57, P. 2550–2555.Dai S.T., Solovyova A. and Winkler P. Calculations of properties of screened He-like systems using correlated wave functions. Phys. Rev. E, 2001, 64, P. 016408(1–9).Dai S.T., Solovyova A. and Winkler P. Calculations of properties of screened He-like systems using correlated wave functions. Phys. Rev. E, 2001, 64, P. 016408(1–9).Accad Y., Pekeris C.L. and Schiff B. S and P states of the Helium isoelectronic sequence up to Z = 10. Phys. Rev. A., 1971, 4, P. 516–536.Accad Y., Pekeris C.L. and Schiff B. S and P states of the Helium isoelectronic sequence up to Z = 10. Phys. Rev. A., 1971, 4, P. 516–536.Katriel J. The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns. J. Chem. Phys., 2012, 136, P. 144112(1–8).Katriel J. The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns. J. Chem. Phys., 2012, 136, P. 144112(1–8).Knight R.E. and Scherr C.W. Two-electron atoms II. A perturbation study of some excited states. Rev. Mod. Phys., 1963, 35, P. 431–436.Knight R.E. and Scherr C.W. Two-electron atoms II. A perturbation study of some excited states. Rev. Mod. Phys., 1963, 35, P. 431–436.Sanders F.C. and Scherr C.W. Perturbation study of some excited states of two-electron atoms. Phys. Rev., 1969, 181, P. 84–97.Sanders F.C. and Scherr C.W. Perturbation study of some excited states of two-electron atoms. Phys. Rev., 1969, 181, P. 84–97.Blanchard P. Nonrelativistic-energy Z-expansion coefficients for singly excited S and P states of two-electron ions. Phys. Rev. A, 1976, 13, P. 1698–1701.Blanchard P. Nonrelativistic-energy Z-expansion coefficients for singly excited S and P states of two-electron ions. Phys. Rev. A, 1976, 13, P. 1698–1701.Baker J.D., Freund D.E., Hill R.N. and Morgan III J.D. Radius of convergence and analytic behavior of the 1/Z expansion. Phys. Rev. A, 1990, 41, P. 1247–1273.Baker J.D., Freund D.E., Hill R.N. and Morgan III J.D. Radius of convergence and analytic behavior of the 1/Z expansion. Phys. Rev. A, 1990, 41, P. 1247–1273.Aashamar K., Lyslo G. and Midtdal J. Variational perturbation theory study of some excited states of two-electron atoms. J. Chem. Phys., 1970, 52, P. 3324–3336.Aashamar K., Lyslo G. and Midtdal J. Variational perturbation theory study of some excited states of two-electron atoms. J. Chem. Phys., 1970, 52, P. 3324–3336.Thakkar A.J. and Smith Jr.V.H. Compact and accurate integral-transform wave functions. II. The 21S, 23S, 21P, and 23P states of the helium-like ions from He through Mg10+. Phys. Rev. A, 1977, 15, P. 16–22.Thakkar A.J. and Smith Jr.V.H. Compact and accurate integral-transform wave functions. II. The 21S, 23S, 21P, and 23P states of the helium-like ions from He through Mg10+. Phys. Rev. A, 1977, 15, P. 16–22.Prudente F.V., Costa L.S. and Vianna J.D.M. A study of two-electron quantum dot spectrum using discrete variable representation method. J. Chem. Phys., 2005, 123, P. 224701(1–11).Prudente F.V., Costa L.S. and Vianna J.D.M. A study of two-electron quantum dot spectrum using discrete variable representation method. J. Chem. Phys., 2005, 123, P. 224701(1–11).Lin Y.C., Fang T.K. and Ho Y.K. Quantum entanglement for helium atom in the Debye plasmas. Phys. Plasmas, 2015, 22, P. 032113(1–8).Lin Y.C., Fang T.K. and Ho Y.K. Quantum entanglement for helium atom in the Debye plasmas. Phys. Plasmas, 2015, 22, P. 032113(1–8).

The authors declare that there are no conflicts of interest present.