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

Journal of Physics B Atomic and Molecular Physics

Number of vibrational levels of the molecular hydrogen ion

A V Matvijenko, Л. И. Пономарев

Published Jan 1, 1972• 1 citation
Publisher

Abstract

The variable phase method was used for the calculation of some properties of the H2+ molecular hydrogen ion. It has been found that in the lowest ungerade static potential of the H2+ molecule there exist 2 bound states. The scattering lengths for the lowest gerade and ungerade potentials are ag=-30au, au=7.6*102au. There is no appreciable change in these values due to the adiabatic corrections.

DOI
10.1088/0022-3700/5/1/011

Related papers

Dependence of the scattering length for hydrogen atoms on effective mass

M. J. Jamieson, Cheung, A.S.C., Henni Ouerdane

The possibility that the non-adiabatic correction to the scattering length of a pair of hydrogen atoms interacting via the ground state molecular potential, X1Σ+g , of H2 could be made by replacing the mass of each nucleus by a different effective mass is explored. The Born-Oppenheimer potential with adiabatic, relativistic and radiative corrections is used in calculations of the scattering lengths and the mass-dependent shifts of the rotationless vibrational levels with fixed and varying effective masses. The shifts are compared with established values and it is demonstrated that the semi-classical formula for the scattering length accounts well for the effect of changing the mass. A perturbing potential that is equivalent to a change in mass is derived and it is compared to a published local non-adiabatic potential.

•Jan 1, 2010•0 citations

Hyperfine-interaction-induced g/u mixing and its implication on the existence of the first excited vibrational level of the A+ Σu+2 state of H2+ and on the scattering length of the H + H+ collision

Maximilian Beyer, Frédéric Merkt

Ab initio calculations of the energy level structure of H2+ that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 − 2) and (v = 1, N = 0)] of the first electronically excited (A+ Σu+2) state of H2+, the (v = 1, N = 0) level having a calculated binding energy of only Eb = 1.082 219 8(4)·10−9 Eh and leading to an extremely large scattering length of 750(5) a0 for the H+ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states (X+ Σg+2 and A+ Σu+2) of H2+ induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H2+, and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A+ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H+ + H(1s) collision are −578(6) a0 and −43(4) a0 for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A+ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H+ + H(1s, F = 1) collision is 757(7) a0.

The Journal of Chemical Physics•Dec 3, 2018•10 citations

Quantization rule for highly excited vibrational states of H

Alexander Kaiser, Tim-Oliver Müller, Harald Friedrich

We analyse the influence of the attractive tail of the p–H two-body potential on the energy progression of the near-threshold rotationless bound states of the H molecular ion. For the 1sσ g configuration we study different definitions of the reference potential describing the potential tail. In each case, the scattering length consistent with the energies of the highly excited vibrational states, as calculated by Hilico et al. [Eur. Phys. J. D 12, 449 (2000)], is found to lie in the range between −30.5 and −31 au. In the 2pσ u configuration, for which there is an extremely weakly bound v = 1 vibrational state [Carbonell et al., Europhys. Lett. 64, 316 (2003)], we obtain a scattering length near 762.8 au. In contrast to previously conveyed impressions, the weakly bound v = 1 state can be satisfactorily described as the excited state in a distant shallow potential well of the p–H two-body potential, provided that a sufficient number of terms is included in the representation of the large-distance behaviour of this potential.

Molecular Physics•Nov 21, 2012•7 citations

Long range adiabatic potentials and scattering lengths for theEF, eandhstates of the hydrogen molecule

Tadeusz Orlikowski, Grażyna Staszewska, L. Wolniewicz

Accurate EF 1Σ g , e 3Σ u and h 3Σ g state adiabatic potentials are computed for internuclear distances up to 44 atomic units. All the energies are more accurate by a fraction of a wave number from existing results. Scattering lengths for collisions of 1S and 2S hydrogen atoms are evaluated. These differ significantly from earlier results. This is because for some states the scattering length is very sensitive to even small changes in the potential.

Molecular Physics•May 20, 1999•43 citations

Ab initio potential-energy curves, scattering lengths, and rovibrational levels of the He2+ molecular ion in excited electronic states

Jacek Gȩbala, Michał Przybytek, Marcin Gronowski +1 more

We calculate accurate potential-energy curves for a ground-state ${\mathrm{He}}^{+}$ ion interacting with a He atom in the lowest-energy metastable $^{3}S$ electronic state. We employ the full configuration-interaction method, equivalent to exact diagonalization, with results extrapolated to the complete basis set limit. The leading relativistic and adiabatic corrections are included using perturbation theory. We calculate rovibrational levels and spectroscopic constants of the ${\mathrm{He}}_{2}^{+}$ molecular ion in excited electronic states for three stable isotopologues. We predict the scattering lengths for ultracold ion-atom collisions. The theoretical data are presented with their uncertainties and agree well with previous results for the ground state. The reported results may be useful for the spectroscopy of the ${\mathrm{He}}_{2}^{+}$ molecular ion in the excited electronic state and collisional studies of ${\mathrm{He}}^{+}$ ions immersed in ultracold gases of metastable He atoms.

arXiv:2301.11307•Physical Review A•Nov 21, 2023•7 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
The variable phase method was used for the calculation of some properties of the H2+ molecular hydrogen ion. It has been found that in the lowest ungerade static potential of the H2+ molecule there exist 2 bound states. The scattering lengths for the lowest gerade and ungerade potentials are ag=-30au, au=7.6*102au. There is no appreciable change in these values due to the adiabatic corrections.
The possibility that the non-adiabatic correction to the scattering length of a pair of hydrogen atoms interacting via the ground state molecular potential, X1Σ+g , of H2 could be made by replacing the mass of each nucleus by a different effective mass is explored. The Born-Oppenheimer potential with adiabatic, relativistic and radiative corrections is used in calculations of the scattering lengths and the mass-dependent shifts of the rotationless vibrational levels with fixed and varying effective masses. The shifts are compared with established values and it is demonstrated that the semi-classical formula for the scattering length accounts well for the effect of changing the mass. A perturbing potential that is equivalent to a change in mass is derived and it is compared to a published local non-adiabatic potential.
Ab initio calculations of the energy level structure of H2+ that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 − 2) and (v = 1, N = 0)] of the first electronically excited (A+ Σu+2) state of H2+, the (v = 1, N = 0) level having a calculated binding energy of only Eb = 1.082 219 8(4)·10−9 Eh and leading to an extremely large scattering length of 750(5) a0 for the H+ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states (X+ Σg+2 and A+ Σu+2) of H2+ induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H2+, and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A+ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H+ + H(1s) collision are −578(6) a0 and −43(4) a0 for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A+ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H+ + H(1s, F = 1) collision is 757(7) a0.
We analyse the influence of the attractive tail of the p–H two-body potential on the energy progression of the near-threshold rotationless bound states of the H molecular ion. For the 1sσ g configuration we study different definitions of the reference potential describing the potential tail. In each case, the scattering length consistent with the energies of the highly excited vibrational states, as calculated by Hilico et al. [Eur. Phys. J. D 12, 449 (2000)], is found to lie in the range between −30.5 and −31 au. In the 2pσ u configuration, for which there is an extremely weakly bound v = 1 vibrational state [Carbonell et al., Europhys. Lett. 64, 316 (2003)], we obtain a scattering length near 762.8 au. In contrast to previously conveyed impressions, the weakly bound v = 1 state can be satisfactorily described as the excited state in a distant shallow potential well of the p–H two-body potential, provided that a sufficient number of terms is included in the representation of the large-distance behaviour of this potential.
We calculate accurate potential-energy curves for a ground-state He+{\mathrm{He}}^{+}He+ ion interacting with a He atom in the lowest-energy metastable 3S^{3}S3S electronic state. We employ the full configuration-interaction method, equivalent to exact diagonalization, with results extrapolated to the complete basis set limit. The leading relativistic and adiabatic corrections are included using perturbation theory. We calculate rovibrational levels and spectroscopic constants of the He2+{\mathrm{He}}_{2}^{+}He2+​ molecular ion in excited electronic states for three stable isotopologues. We predict the scattering lengths for ultracold ion-atom collisions. The theoretical data are presented with their uncertainties and agree well with previous results for the ground state. The reported results may be useful for the spectroscopy of the He2+{\mathrm{He}}_{2}^{+}He2+​ molecular ion in the excited electronic state and collisional studies of He+{\mathrm{He}}^{+}He+ ions immersed in ultracold gases of metastable He atoms.
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