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

Revista Mexicana de Física

The stability of the $1u$ state of H₂⁺ in magnetic fields with arbitrary orientations

Daniel Julian Nader, Juan Carlos Lopez Vieyra

Published Jan 1, 2016• 0 citations
Publisher

Abstract

The existence and stability of the van der Waals $1u$ 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 $0 \le B \le 10\,$ a.u. (1 a.u. $\equiv 2.35\times10^{9}\,$G) 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 $0 \leq B \lesssim 6.6$ a.u. where the $1u$ state in parallel configuration is stable towards dissociation \hbox{\ppe $\nrightarrow$ H + p}. The maximal stability is found to be at magnetic field $B\sim 2\,$a.u. where the potential well supports one vibrational state below the dissociation limit.

DOI
10.31349/revmexfis.62.5.461

Related papers

He2+ molecular ion and the He− atomic ion in strong magnetic fields

Juan Carlos Lopez Vieyra, Alexander V. Turbiner

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 ${{\mathrm{He}}_{2}}^{+}$ 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 ${{\mathrm{He}}_{2}}^{+}$ 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 $B=1000\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, the dissociation energy of ${{\mathrm{He}}_{2}}^{+}$ 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.

arXiv:1704.01118•Physical Review A•Aug 9, 2017•3 citations

H32+ molecular ion in a strong magnetic field: Triangular configuration

Juan Carlos Lopez Vieyra, Alexander V. Turbiner

The existence of the molecular ion ${\mathrm{H}}_{3}^{2+}$ in a magnetic field in a triangular configuration is revised. A variational method with an optimization of the form of the vector potential (gauge fixing) is used. It is shown that in the range of magnetic fields ${10}^{8}<B<{10}^{11}\mathrm{G}$ the system $(\mathrm{pppe}),$ with the protons forming an equilateral triangle perpendicular to the magnetic line, has a well-pronounced minimum in the total energy. This configuration is unstable under the decays $(\mathrm{H}\mathrm{atom})+p+p$ and ${\mathrm{H}}_{2}^{+}+p.$ The triangular configuration of ${\mathrm{H}}_{3}^{2+}$ complements ${\mathrm{H}}_{3}^{2+}$ in the linear configuration that exists for $B\ensuremath{\gtrsim}{10}^{10}\mathrm{G}.$

arXiv:astro-ph/0203020•Physical Review A•Aug 26, 2002•20 citations

The HeH+molecular ion in a magnetic field

A V Turbiner, N L Guevara

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.

arXiv:physics/0703090•Journal of Physics B Atomic Molecular and Optical Physics•Aug 6, 2007•10 citations

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

T. Detmer, Peter Schmelcher, F. Κ. Diakonos +1 more

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.

arXiv:physics/9711020•Physical Review A•Sep 1, 1997•71 citations

He22+ molecular ion can exist in a magnetic field

Alexander V. Turbiner, Nicolais L. Guevara

We present a detailed study of the ground state of the Coulomb system $(\ensuremath{\alpha}\ensuremath{\alpha}ee)$ which corresponds to the ${\mathrm{He}}_{2}^{2+}$ 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.

arXiv:astro-ph/0610928•Physical Review A•Dec 26, 2006•15 citations

Existence of the finite hydrogenic molecular chain H3 and ion H2− in a strong magnetic field

Daniel Julian Nader, Juan Carlos Lopez Vieyra, Alexander V. Turbiner

The existence and stability of the linear hydrogenic chain $\mathrm{H}{}_{3}$ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ in a strong magnetic field is established. Variational calculations for $\mathrm{H}{}_{3}$ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ are carried out in magnetic fields in the range ${10}^{11}\ensuremath{\le}B\ensuremath{\le}{10}^{13}$ G with a 17-parametric (13-parametric for $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$), physically adequate trial function. Protons are assumed to be infinitely massive, fixed along the magnetic line. States with total spin projection ${S}_{z}=\ensuremath{-}3/2$ and magnetic quantum numbers $M=\ensuremath{-}3,\ensuremath{-}4,\ensuremath{-}5$ are studied. It is shown that for both $\mathrm{H}{}_{3}$ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$, the lowest-energy state corresponds to $M=\ensuremath{-}3$ in the whole range of magnetic fields studied. As for a magnetic field $B\ensuremath{\gtrsim}{10}^{11}$ G, both $\mathrm{H}{}_{3}$ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ exist as metastable states, becoming stable for $B\ensuremath{\ge}1.9\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G and for $B\ensuremath{\ge}2.7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G, respectively. The excited states ${}^{4}{(\ensuremath{-}4)}^{+}$, ${}^{4}{(\ensuremath{-}5)}^{+}$ of ${\mathrm{H}}_{3}$ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ appear at magnetic fields $B>7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ and ${10}^{12}$ G, respectively.

arXiv:1903.06533•Physical Review A•Jul 11, 2019•4 citations
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.
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.
The existence of the molecular ion H32+{\mathrm{H}}_{3}^{2+}H32+​ in a magnetic field in a triangular configuration is revised. A variational method with an optimization of the form of the vector potential (gauge fixing) is used. It is shown that in the range of magnetic fields 108<B<1011G{10}^{8}<B<{10}^{11}\mathrm{G}108<B<1011G the system (pppe),(\mathrm{pppe}),(pppe), with the protons forming an equilateral triangle perpendicular to the magnetic line, has a well-pronounced minimum in the total energy. This configuration is unstable under the decays (Hatom)+p+p(\mathrm{H}\mathrm{atom})+p+p(Hatom)+p+p and H2++p.{\mathrm{H}}_{2}^{+}+p.H2+​+p. The triangular configuration of H32+{\mathrm{H}}_{3}^{2+}H32+​ complements H32+{\mathrm{H}}_{3}^{2+}H32+​ in the linear configuration that exists for $B\ensuremath{\gtrsim}{10}^{10}\mathrm{G}.$
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.
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 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.
The existence and stability of the linear hydrogenic chain H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ in a strong magnetic field is established. Variational calculations for H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ are carried out in magnetic fields in the range ${10}^{11}\ensuremath{\le}B\ensuremath{\le}{10}^{13}$ G with a 17-parametric (13-parametric for $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$), physically adequate trial function. Protons are assumed to be infinitely massive, fixed along the magnetic line. States with total spin projection ${S}_{z}=\ensuremath{-}3/2$ and magnetic quantum numbers $M=\ensuremath{-}3,\ensuremath{-}4,\ensuremath{-}5$ are studied. It is shown that for both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$, the lowest-energy state corresponds to $M=\ensuremath{-}3$ in the whole range of magnetic fields studied. As for a magnetic field $B\ensuremath{\gtrsim}{10}^{11}$ G, both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ exist as metastable states, becoming stable for $B\ensuremath{\ge}1.9\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G and for $B\ensuremath{\ge}2.7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G, respectively. The excited states ${}^{4}{(\ensuremath{-}4)}^{+}$, ${}^{4}{(\ensuremath{-}5)}^{+}$ of H3{\mathrm{H}}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ appear at magnetic fields $B>7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ and 1012{10}^{12}1012 G, respectively.