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arXiv:physics/9711020 · Physical Review A

Hydrogen molecule in magnetic fields: The ground states of the Σ manifold of the parallel configuration

T. Detmer, Peter Schmelcher, F. Κ. Diakonos, L. S. Cederbaum

Published Sep 1, 1997• 71 citations
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Abstract

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 $B=0$ up to $B=100\mathrm{a}.\mathrm{u}.$ 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 $B$ 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.

DOI
10.1103/physreva.56.1825

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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\ensuremathΣg{}^{1}{\ensuremath{\Sigma}}_{g}1\ensuremathΣg​ 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\ensuremathΣg{}^{3}{\ensuremath{\Sigma}}_{g}3\ensuremathΣg​ state is shown to develop an additional minimum which has no counterpart in field-free space. The 1\ensuremathΣu{}^{1}{\ensuremath{\Sigma}}_{u}1\ensuremathΣu​ 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 H2\ensuremath→H\ensuremath−+H+{\mathrm{H}}_{2}{\ensuremath{\rightarrow}\mathrm{H}}^{\mathrm{\ensuremath{-}}}{+\mathrm{H}}^{+}H2​\ensuremath→H\ensuremath−+H+ for magnetic field strengths B\ensuremath≳20a.u.B\ensuremath{\gtrsim}20\mathrm{a}.\mathrm{u}.B\ensuremath≳20a.u. due to the existence of strongly bound H\ensuremath−{\mathrm{H}}^{\mathrm{\ensuremath{-}}}H\ensuremath− states in strong magnetic fields. The repulsive 3\ensuremathΣu{}^{3}{\ensuremath{\Sigma}}_{u}3\ensuremathΣu​ 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\ensuremathΣg{}^{1}{\ensuremath{\Sigma}}_{g}1\ensuremathΣg​ and 3\ensuremathΣu{}^{3}{\ensuremath{\Sigma}}_{u}3\ensuremathΣu​ states cross as a function of BBB and the 3\ensuremathΣu{}^{3}{\ensuremath{\Sigma}}_{u}3\ensuremathΣu​ state, which is an unbound state, becomes the ground state of the hydrogen molecule in magnetic fields B\ensuremath≳0.2a.u.B\ensuremath{\gtrsim}0.2\mathrm{a}.\mathrm{u}.B\ensuremath≳0.2a.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\ensuremathΠu{}^{3}{\ensuremath{\Pi}}_{u}3\ensuremathΠu​ state.
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 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.
A detailed study of the low-lying electronic states 1 Σ, 3 Σ, 3 Π, 3 Δ of the HeH + molecular ion in parallel to a magnetic field configuration (when α-particle and proton are situated on the same magnetic line) is carried out for B = 0 − 4.414 × 10 13 G in the Born–Oppenheimer approximation. The variational method is employed using a physically adequate trial function. It is shown that for the Σ states the parallel configuration is stable with respect to small deviations. The quantum numbers of the ground state depend on the magnetic field strength. The ground state evolves from the spin-singlet 1 Σ state for small magnetic fields B ≲ 0.5 au to the spin-triplet 3 Σ (unbound or weakly bound) state for intermediate fields and to the spin-triplet strongly bound 3 Π state for B ≳ 15 au. When the HeH + molecular ion exists, it is stable with respect to dissociation.
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.
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.
The ground states of small atoms and molecules, namely those of the hydrogen, helium, and lithium atoms and the hydrogen molecular ion and hydrogen molecule, are studied in the presence of an external magnetic field. For the one-electron systems the ground state is of the same symmetry for arbitrary field strengths, and the corresponding binding energy shows a monotonic increase with increasing field strength. More interesting, the two- and three-electron systems show ground-state transitions which involve both transitions in the spin multiplicity and spatial quantum numbers. An outlook on the expected cascade of transitions in many electron atoms and molecules is provided. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 789–795, 1998