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arXiv:physics/0703090 · Journal of Physics B Atomic Molecular and Optical Physics

The HeH+molecular ion in a magnetic field

A V Turbiner, N L Guevara

Published Aug 6, 2007• 10 citations
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Abstract

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.

DOI
10.1088/0953-4075/40/16/006

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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.
We present a detailed study of the ground state of the Coulomb system $(\ensuremath{\alpha}\ensuremath{\alpha}ee)$ which corresponds to the He22+{\mathrm{He}}_{2}^{2+}He22+​ molecular ion in a magnetic field $B=0\ensuremath{-}4.414\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$ in parallel configuration (it is assumed that infinitely massive $\ensuremath{\alpha}$ particles are situated along a magnetic-field line). One uses a variational method, introducing a trial function which includes electronic correlation in the form $\mathrm{exp}(\ensuremath{\gamma}{r}_{12})$, where $\ensuremath{\gamma}$ is a variational parameter. It is shown that the quantum numbers of the lowest total-energy state depend on the magnetic-field strength. This state evolves from the spin-singlet $^{1}{\ensuremath{\Sigma}}_{g}$ metastable state at $0\ensuremath{\le}B\ensuremath{\lesssim}0.85\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ to a repulsive spin-triplet $^{3}{\ensuremath{\Sigma}}_{u}$ state for $0.85\ensuremath{\lesssim}B\ensuremath{\lesssim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ and, finally, to a stable strongly bound spin-triplet $^{3}{\ensuremath{\Pi}}_{u}$ state at stronger fields $B\ensuremath{\gtrsim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ The lowest vibrational energy for the latter case is calculated.
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.
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 one-electron lithium-containing Coulomb systems of atomic type Li 2 + and molecular type Li 5 + 2 , LiHe 4 + and LiH 3 + are studied in the presence of a strong magnetic field B ⩽ 10 7 au in a non-relativistic framework. They are considered at the Born–Oppenheimer approximation of zero order (infinitely massive centres) within the parallel configuration (molecular axis parallel to the magnetic field). The variational and Lagrange-mesh methods are employed, complementing each other. It is demonstrated that the molecular systems LiH 3 + , LiHe 4 + and Li 5 + 2 can exist for sufficiently strong magnetic fields B ≳ 10 4 au and that Li 5 + 2 can even be stable at magnetic fields typical of magnetars.
The stability of the 1u1u1u state of H₂⁺ in magnetic fields with arbitrary orientations
The existence and stability of the van der Waals 1u1u1u state of the molecular ion \ppe with fixed centers (infinitely massive nuclei) in the presence of a magnetic field with arbitrary orientations in the range of strengths 0≤B≤10 0 \le B \le 10\,0≤B≤10 a.u. (1 a.u. ≡2.35×109 \equiv 2.35\times10^{9}\,≡2.35×109G) is studied within the non-relativistic framework. The study is based on the variational method with physical relevant trial functions. A particular emphasis to the gauge optimization through variational parameters is incorporated. It is shown that, for all the magnetic fields studied, the potential energy curve has a pronounced minimum for finite internuclear distances and the optimal configuration of minimal total energy is realized when the molecular axis is oriented along the magnetic field lines (parallel configuration). We found a domain of magnetic fields 0≤B≲6.60 \leq B \lesssim 6.60≤B≲6.6 a.u. where the 1u1u1u state in parallel configuration is stable towards dissociation \hbox{\ppe ↛\nrightarrow↛ H + p}. The maximal stability is found to be at magnetic field B∼2 B\sim 2\,B∼2a.u. where the potential well supports one vibrational state below the dissociation limit.