Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

H− and He in a uniform magnetic field: Ground-state wave functions, energies, and binding energies for fields below 109 G

Chang‐Hwan Park, Anthony F. Strace

Published Feb 1, 1984• 40 citations
Publisher

Abstract

Wave functions, energies, and binding energies for the lowest singlet states of ${\mathrm{H}}^{\ensuremath{-}}$ and He in uniform magnetic fields $B<{10}^{9}$ G are calculated using an adiabatic approximation in hyperspherical coordinates. In computing the angular part of the wave functions, a coupled expansion in one-electron oblate spheroidal angle functions is used. In addition to contracting the two-electron wave function radially, the magnetic field is found to distort the angular part of the wave function mainly by reducing the $\mathrm{ss}^{1}S$ character of the state and replacing it with an $\mathrm{sd}^{1}D$ character. Results for energies and binding energies are comparable with those obtained in variational calculations. In order to compute the binding energies of ${\mathrm{H}}^{\ensuremath{-}}$ and He we have also calculated the binding energies of the ground states of H and ${\mathrm{He}}^{+}$ in uniform magnetic fields $B<{10}^{9}$ G using the adiabatic oblate-spheroidal-coordinate method of Starace and Webster.

DOI
10.1103/physreva.29.442

Related papers

H- and He in a uniform magnetic field: Ground-state wave functions, energies, and binding energies for fields below 109 G

Chang‐Hwan Park, Anthony F. Starace

Wave functions, energies, and binding energies for the lowest singlet states of H- and He in uniform magnetic fields B<109 G are calculated using an adiabatic approximation in hyperspherical coordinates. In computing the angular part of the wave functions, a coupled expansion in one-electron oblate spheroidal angle functions is used. In addition to contracting the two-electron wave function radially, the magnetic field is found to distort the angular part of the wave function mainly by reducing the ss1S character of the state and replacing it with an sd 1D character. Results for energies and binding energies are comparable with those obtained in variational calculations. In order to compute the binding energies of H- and He we have also calculated the binding energies of the ground states of H and He+ in uniform magnetic fields B<109 G using the adiabatic oblate-spheroidal-coordinate method of Starace and Webster.

•Feb 1, 1984•0 citations

Binding energy and triplet-singlet splitting for the hydrogen molecule in ultrahigh magnetic fields

A. V. Korolev, M. A. Liberman

The energy difference of triplet and singlet states due to exchange coupling, the energy levels, the ground binding state, and the wave functions is calculated for a hydrogen molecule in an ultrahigh magnetic field when the distance between the Landau levels e\ensuremath{\Elzxh}H/mc exceeds the Coulomb unit (Rydberg) ${\mathit{me}}^{4}$/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$. The results are asymptotically exact at the interatomic distance R, which is large compared with the atomic size and for high magnetic fields. It is shown here that the triplet electron term has a minimum at large interatomic distances corresponding to the formation of the ground state of a stable ${\mathrm{H}}_{2}$ molecule. For the hydrogen molecule in an ultrahigh magnetic field, the energy of the singlet state also has a deep minimum at R1, which is much higher than the triplet one and may be a metastable state only.

Physical Review A•Feb 1, 1992•44 citations

Hydrogen and helium atoms in strong magnetic fields

Anand Thirumalai, Jeremy Heyl

The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the binding energies of the first few low-lying states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these one- and two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.

arXiv:0806.3113•Physical Review A•Jan 30, 2009•38 citations

Correlation Energy of an Electron Gas in the Quantum Strong-Field Limit

Gerd E. Keiser, Fa Yueh Wu

The ground state of an electron gas in an intense magnetic field is studied using a wave function of the product form ${\ensuremath{\Psi}}_{0}^{B}\ensuremath{\Phi}$. The correlation factor ${\ensuremath{\Psi}}_{0}^{B}$ is taken to be the ground-state solution of a charged Bose gas and $\ensuremath{\Phi}$ a determinant of single-particle Landau states. In the quantum strong-field limit so that only the lowest Landau state is populated, the correlation energy is computed using the cluster-expansion technique and a variational determination of the boson energy. Numerical results obtained are lower than those derived under the random-phase approximation.

Physical Review A•Oct 1, 1973•9 citations

Calculations of the binding energies of weakly bound He-He-H, He-He-H-and He-H-H molecules

Yong Li, C. D. Lin

We searched for the existence of weakly bound He-He-H, He-He-H - and He-H-H molecules where the binding energies are expected to be of the order of degrees Kelvin. By employing the best empirical interaction between each pair of particles, we solved the Schrödinger equation for the triatomic systems using hyperspherical coordinates in the adiabatic approximation. We found no bound states for 4 He- 4 He- 1 H, three bound states for 4 He- 4 He- 1 H - and one bound state for 4 He- 1 H- 1 H. Bound states for other isotope combinations are also examined.

Journal of Physics B Atomic Molecular and Optical Physics•Oct 7, 1999•22 citations

Hydrogen Atom and Hydrogen Molecule Ion in Homogeneous Magnetic Fields of Arbitrary Strength

Ranjan Bhaduri, Yozo Nogami, Chindhu S. Warke

A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics

The Astrophysical Journal•Oct 1, 1977•36 citations
Wave functions, energies, and binding energies for the lowest singlet states of H\ensuremath−{\mathrm{H}}^{\ensuremath{-}}H\ensuremath− and He in uniform magnetic fields B<109B<{10}^{9}B<109 G are calculated using an adiabatic approximation in hyperspherical coordinates. In computing the angular part of the wave functions, a coupled expansion in one-electron oblate spheroidal angle functions is used. In addition to contracting the two-electron wave function radially, the magnetic field is found to distort the angular part of the wave function mainly by reducing the ss1S\mathrm{ss}^{1}Sss1S character of the state and replacing it with an sd1D\mathrm{sd}^{1}Dsd1D character. Results for energies and binding energies are comparable with those obtained in variational calculations. In order to compute the binding energies of H\ensuremath−{\mathrm{H}}^{\ensuremath{-}}H\ensuremath− and He we have also calculated the binding energies of the ground states of H and He+{\mathrm{He}}^{+}He+ in uniform magnetic fields B<109B<{10}^{9}B<109 G using the adiabatic oblate-spheroidal-coordinate method of Starace and Webster.
Wave functions, energies, and binding energies for the lowest singlet states of H- and He in uniform magnetic fields B<109 G are calculated using an adiabatic approximation in hyperspherical coordinates. In computing the angular part of the wave functions, a coupled expansion in one-electron oblate spheroidal angle functions is used. In addition to contracting the two-electron wave function radially, the magnetic field is found to distort the angular part of the wave function mainly by reducing the ss1S character of the state and replacing it with an sd 1D character. Results for energies and binding energies are comparable with those obtained in variational calculations. In order to compute the binding energies of H- and He we have also calculated the binding energies of the ground states of H and He+ in uniform magnetic fields B<109 G using the adiabatic oblate-spheroidal-coordinate method of Starace and Webster.
The energy difference of triplet and singlet states due to exchange coupling, the energy levels, the ground binding state, and the wave functions is calculated for a hydrogen molecule in an ultrahigh magnetic field when the distance between the Landau levels e\ensuremath{\Elzxh}H/mc exceeds the Coulomb unit (Rydberg) me4{\mathit{me}}^{4}me4/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$. The results are asymptotically exact at the interatomic distance R, which is large compared with the atomic size and for high magnetic fields. It is shown here that the triplet electron term has a minimum at large interatomic distances corresponding to the formation of the ground state of a stable H2{\mathrm{H}}_{2}H2​ molecule. For the hydrogen molecule in an ultrahigh magnetic field, the energy of the singlet state also has a deep minimum at R1, which is much higher than the triplet one and may be a metastable state only.
The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the binding energies of the first few low-lying states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these one- and two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.
The ground state of an electron gas in an intense magnetic field is studied using a wave function of the product form ${\ensuremath{\Psi}}_{0}^{B}\ensuremath{\Phi}$. The correlation factor ${\ensuremath{\Psi}}_{0}^{B}$ is taken to be the ground-state solution of a charged Bose gas and $\ensuremath{\Phi}$ a determinant of single-particle Landau states. In the quantum strong-field limit so that only the lowest Landau state is populated, the correlation energy is computed using the cluster-expansion technique and a variational determination of the boson energy. Numerical results obtained are lower than those derived under the random-phase approximation.
We searched for the existence of weakly bound He-He-H, He-He-H - and He-H-H molecules where the binding energies are expected to be of the order of degrees Kelvin. By employing the best empirical interaction between each pair of particles, we solved the Schrödinger equation for the triatomic systems using hyperspherical coordinates in the adiabatic approximation. We found no bound states for 4 He- 4 He- 1 H, three bound states for 4 He- 4 He- 1 H - and one bound state for 4 He- 1 H- 1 H. Bound states for other isotope combinations are also examined.
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics