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Physica Scripta

Terms of a Hydrogen Molecule in a High Magnetic Field

M. A. Liberman, A V Petrov

Published May 1, 1998• 3 citations
Publisher

Abstract

Analytical expressions for the pair interaction potential between hydrogen atoms in a high magnetic field are derived for the triplet and singlet terms. The results are asymptotically exact for large interatomic distance R compared with the atomic size and in the limit of ultrahigh magnetic fields. It is shown that the ground state of the hydrogen molecule in a high magnetic field is the triplet term 3 Π u , which corresponds to the case when one atom is in the ground state and the other is in the first excited state. The interaction between two ground-state hydrogen atoms in the term 3 Σ u is very weak. The energy of the singlet term 1 Σ g has a deep minimum, which is much higher than the triplet one and may be a metastable state of a hydrogen molecule only.

DOI
10.1088/0031-8949/57/5/005

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Ground state of the hydrogen molecule in a strong magnetic field

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The ground state of a hydrogen molecule in magnetic fields is investigated using a fully numerical Hartree-Fock approach. We found that between $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state is ${}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ with very weak interaction between atoms. In this field region the hydrogen can form a superfluid phase, predicted earlier [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]; the state ${}^{3}{\ensuremath{\Pi}}_{u}$ is metastable and may be responsible for the unknown excitonic line observed by Timofeev and Chernenko [JETP Lett. 61, 617 (1995)]. For magnetic fields stronger than $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state is the tightly bound ${}^{3}{\ensuremath{\Pi}}_{u}.$

Physical Review A•Oct 1, 1997•25 citations
Analytical expressions for the pair interaction potential between hydrogen atoms in a high magnetic field are derived for the triplet and singlet terms. The results are asymptotically exact for large interatomic distance R compared with the atomic size and in the limit of ultrahigh magnetic fields. It is shown that the ground state of the hydrogen molecule in a high magnetic field is the triplet term 3 Π u , which corresponds to the case when one atom is in the ground state and the other is in the first excited state. The interaction between two ground-state hydrogen atoms in the term 3 Σ u is very weak. The energy of the singlet term 1 Σ g has a deep minimum, which is much higher than the triplet one and may be a metastable state of a hydrogen molecule only.
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.
We investigate the hydrogen molecule in a strong parallel magnetic field using a fully numerical Hartree-Fock approach. We find that for magnetic fields below $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T the ground state of H2{\mathrm{H}}_{2}H2​ is the strongly bound singlet state ${}^{1}{\ensuremath{\Sigma}}_{g},$ for magnetic fields stronger than $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state of the molecule is the strongly bound triplet ${}^{3}{\ensuremath{\Pi}}_{u},$ and for magnetic fields between $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the symmetry of the ground state is the triplet state ${}^{3}{\ensuremath{\Sigma}}_{u},$ which is characterized by repulsion at intermediate internuclear distances and by a weak quadrupole-quadrupole interaction between atoms at large internuclear separation. In this region of magnetic field strength the hydrogen molecule is bound weakly, if at all; the hydrogen atoms behave like a weakly nonideal gas of Bose particles and can form a superfluid phase predicted in earlier works [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]. For magnetic fields between $\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the triplet state ${}^{3}{\ensuremath{\Pi}}_{u}$ is found to be metastable. This state may be responsible for an unknown excitonic line observed experimentally [Timofeev and Chernenko, JETP Lett. 61, 617 (1995)].
The electronic structure of the hydrogen molecule is investigated for the parallel configuration. The ground states of the \ensuremath{\Sigma} manifold are studied for ungerade and gerade parity as well as singlet and triplet states covering a broad regime of field strengths from B=0B=0B=0 up to B=100a.u.B=100\mathrm{a}.\mathrm{u}.B=100a.u. A variety of interesting phenomena can be observed. For the ${}^{1}{\ensuremath{\Sigma}}_{g}$ state we found a monotonous decrease of the equilibrium distance and a simultaneous increase of the dissociation energy with growing magnetic-field strength. The ${}^{3}{\ensuremath{\Sigma}}_{g}$ state is shown to develop an additional minimum which has no counterpart in field-free space. The ${}^{1}{\ensuremath{\Sigma}}_{u}$ state shows a monotonous increase in the dissociation energy with a first increasing and then decreasing internuclear distance of the minimum. For this state the dissociation channel is ${\mathrm{H}}_{2}{\ensuremath{\rightarrow}\mathrm{H}}^{\mathrm{\ensuremath{-}}}{+\mathrm{H}}^{+}$ for magnetic field strengths $B\ensuremath{\gtrsim}20\mathrm{a}.\mathrm{u}.$ due to the existence of strongly bound ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ states in strong magnetic fields. The repulsive ${}^{3}{\ensuremath{\Sigma}}_{u}$ state possesses a very shallow van der Waals minimum for magnetic-field strengths smaller than 1.0 a.u. within the numerical accuracy of our calculations. The ${}^{1}{\ensuremath{\Sigma}}_{g}$ and ${}^{3}{\ensuremath{\Sigma}}_{u}$ states cross as a function of BBB and the ${}^{3}{\ensuremath{\Sigma}}_{u}$ state, which is an unbound state, becomes the ground state of the hydrogen molecule in magnetic fields $B\ensuremath{\gtrsim}0.2\mathrm{a}.\mathrm{u}.$ This is of particular interest for the existence of molecular hydrogen in the vicinity of white dwarfs. In superstrong fields the ground state is again a strongly bound state, the ${}^{3}{\ensuremath{\Pi}}_{u}$ state.
We study the ground-state structural properties of a hydrogen molecule in superstrong magnetic fields (B\ensuremath{\approxeq}1012{10}^{12}1012 G) using quantum Monte Carlo (fixed-phase and variational) approaches. We determine that the ground state (spin-triplet) belongs to the sector of total (z-component) angular momentum M=-1 (3{(}^{3}(3${\mathrm{\ensuremath{\Pi}}}_{0}$), meaning that paramagnetic contributions to the total energy cannot be neglected. This non-time-reversal invariant ground state has a strong interatomic interaction, suggesting that a hydrogen gas under the same physical conditions has a tendency to form strong bonded molecules.
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 ground state of a hydrogen molecule in magnetic fields is investigated using a fully numerical Hartree-Fock approach. We found that between $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state is ${}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ with very weak interaction between atoms. In this field region the hydrogen can form a superfluid phase, predicted earlier [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]; the state ${}^{3}{\ensuremath{\Pi}}_{u}$ is metastable and may be responsible for the unknown excitonic line observed by Timofeev and Chernenko [JETP Lett. 61, 617 (1995)]. For magnetic fields stronger than $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state is the tightly bound ${}^{3}{\ensuremath{\Pi}}_{u}.$