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arXiv:math-ph/0403034 · Journal of Physics B Atomic Molecular and Optical Physics

H+2in a strong magnetic field described via a solvable model

Rafael D. Benguria, Raymond Brummelhuis, Pierre Duclos, S. Pérez-Oyarzún

Published May 18, 2004• 17 citations
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Abstract

We consider the hydrogen molecular ion $H^+_2$ in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strenght of the magnnetic field. However we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion, can be obtained when incorporating perturbative corrections up to second order. Finally, we show that $ He_2^{3+}$ exists for sufficiently large magnetic fields.

DOI
10.1088/0953-4075/37/11/008

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Ground states of atoms and molecules in strong magnetic fields

P. Schmelcher

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

International Journal of Quantum Chemistry•Jan 1, 1998•5 citations
We consider the hydrogen molecular ion H2+H^+_2H2+​ in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strenght of the magnnetic field. However we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion, can be obtained when incorporating perturbative corrections up to second order. Finally, we show that He23+ He_2^{3+}He23+​ exists for sufficiently large magnetic fields.
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.
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 study the question of existence, i.e., stability with respect to dissociation of the spin-quartet permutation- and reflection-symmetric ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ (${S}_{z}=\ensuremath{-}3/2,M=\ensuremath{-}3$) state of the $(\ensuremath{\alpha}\ensuremath{\alpha}eee)$ Coulomb system: the He2+{{\mathrm{He}}_{2}}^{+}He2​+ molecular ion, placed in a magnetic field $0\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. We assume that the $\ensuremath{\alpha}$ particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several helium-containing compounds in strong magnetic fields, in particular; we study the spin-quartet ground state of the ${\mathrm{He}}^{\ensuremath{-}}$ ion and the ground (spin-triplet) state of the helium atom, both for a magnetic field in $100\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. The main result is that the He2+{{\mathrm{He}}_{2}}^{+}He2​+ molecular ion in the state ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ is stable towards all possible decay modes for magnetic fields $B\ensuremath{\gtrsim}120\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$ and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At B=1000a.u.B=1000\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.B=1000a.u., the dissociation energy of He2+{{\mathrm{He}}_{2}}^{+}He2​+ into ${\mathrm{He}}^{\ensuremath{-}}+\ensuremath{\alpha}$ is $\ensuremath{\sim}702\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ and the dissociation energy for the decay channel to $\mathrm{He}+\ensuremath{\alpha}+e$ is $\ensuremath{\sim}729\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$, and both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.
EXOTIC MOLECULAR IONS ( He H )2+ AND He23+{\rm He}_2^{3+}He23+​ IN A STRONG MAGNETIC FIELD: LOW-LYING STATES
The Coulombic systems (αpe) and (ααe), (αppe), (ααpe) and ( Li 3+ Li 3+e) placed in a magnetic field B ≳ 1011 G are studied. It is demonstrated a theoretical existence of the exotic ion ( He H )2+ for B ≳ 5 × 1012 G in parallel configuration (the magnetic field is directed along internuclear axis) as optimal as well as its excited states 1π, 1δ. As for the exotic ion [Formula: see text] it is shown that in spite of strong electrostatic repulsion of α-particles this ion can also exist for B ≳ 100 a.u. (= 2.35 × 1011 G ) in parallel configuration as optimal in the states 1σg (ground state), 1πu, 1δg. Upon appearance both ions are unstable towards dissociation with He + in the final state but with very large lifetime. However, at B ≳ 10000 a.u. , the ion ( He H )2+ becomes stable, while at B ≳ 1000 a.u. , the ion [Formula: see text] becomes stable. For both ions the vibrational and rotational energies are calculated. With a magnetic field growth, both exotic ions become more and more tightly bound and compact, their lowest rotational and vibrational energies grow drastically. At the edge of applicability of nonrelativistic approximation, B ~ 4.414 × 1013 G , there are indications that three more exotic linear ions ( H – He – H )3+, ( He – H – He )4+ and even [Formula: see text] in parallel configuration may also occur.
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 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