Finite Hydrogenic molecular chain H$_3$ and ion H$_2^-$ exist in a strong magnetic field

TL;DR

This study demonstrates that the hydrogenic chains H₃ and H₂⁻ can exist and become stable in strong magnetic fields, with specific states becoming stable above certain magnetic field thresholds, using variational calculations.

Contribution

The paper provides the first variational analysis of H₃ and H₂⁻ chains in magnetic fields up to 10¹³ G, establishing their existence and stability thresholds.

Findings

H₃ and H₂⁻ exist as metastable states at B > 10¹¹ G.

H₃ stabilizes at B ≥ 1.9×10¹¹ G.

H₂⁻ stabilizes at B ≥ 2.7×10¹¹ G.

Abstract

The existence and stability of the linear hydrogenic chain H$_3$ and H${}_2^-$ in a strong magnetic field is established. Variational calculations for H$_3$ and H${}_2^-$ are carried out in magnetic fields in the range $10^{11}\leq B \leq 10^{13}\,$G with 17-parametric (13-parametric for H${}_2^-$), physically adequate trial function. Protons are assumed infinitely massive, fixed along the magnetic line. States with total spin projection $S_z=-3/2$ and magnetic quantum numbers $M=-3,-4,-5$ are studied. It is shown that for both H$_3$ and H${}_2^-$ the lowest energy state corresponds to $M=-3$ in the whole range of magnetic fields studied. As for a magnetic field $B \gtrsim 10^{11}\,$G both H$_3$ and H${}_2^-$ exist as metastable states, becoming stable for $B \geq 1.9 \times 10^{11}\,$G and for $B \geq 2.7 \times 10^{11}\,$G, respectively. The excited states $^4(-4)^+$, $^4(-5)^+$ of ${\rm H}_3$ and H${}_2^-$ appear at magnetic fields $B > 7 \times 10^{11}$ and $10^{12}$\,G, respectively.

Tables

Table 1. Table 1: Ground state of H 3 subscript H 3 {\rm H}_{3} : total energy E T subscript 𝐸 𝑇 E_{T} and equilibrium distance R e ​ q subscript 𝑅 𝑒 𝑞 R_{eq} in symmetric configuration ( R + = R − ) subscript 𝑅 subscript 𝑅 (R_{+}=R_{-}) vs magnetic field for the ground state ( − 3 ) + 4 {}^{4}(-3)^{+} . Results Lai-Salpeter:1992 marked by † † \dagger as for results for H 2 Turbiner:2010 , H + 3 superscript subscript absent 3 {}_{3}^{+} Turbiner:2007 and H Kravchenko:1996 . Lowest longitudinal vibrational energies for (anti)symmetrical ( s ) 𝑠 (s) and ( a ) 𝑎 (a) modes presented as well as dissociation energies into two decay channels. Energies and distances in a.u.

$B\left({10}^{9}G\right)$	$E_T^{H_3}$	$R_{eq}^{H_3}$	$E_{vib}^{0\left(s\right)}$	$E_{vib}^{0\left(a\right)}$	$E_{diss}^{H_3\to H_2+H}$	$E_{diss}^{H_3\to H_3^{+}+e}$
$100.0$	-8.62	0.482			-	-
	-8.50${}^{†}$	0.48
$117.5$	-9.18	0.460			-0.174	-
$235.0$	-12.09	0.359	0.073	0.150	0.127	2.63
$500.0$	-16.15	0.277	0.104	0.208	0.644	-
	-16.08${}^{†}$	0.27${}^{†}$
$1000.0$	-20.90	0.219	0.154	0.282	1.382	-
	-20.81${}^{†}$	0.22${}^{†}$
$2350.0$	-28.39	0.168	0.237	0.425	2.759	6.12
$5000.0$	-36.78	0.134	0.337	0.601	-	-
	-36.77${}^{†}$	0.13${}^{†}$
$10000.0$	-46.19	0.110	0.467	0.814	-	-
	-46.19${}^{†}$	0.11${}^{†}$

Table 2. Table 2: Ground state H 2 − superscript subscript H 2 {\rm H}_{2}^{-} : total energy E T subscript 𝐸 𝑇 E_{T} and equilibrium distance R e ​ q subscript 𝑅 𝑒 𝑞 R_{eq} for the state ( − 3 ) + 4 {}^{4}(-3)^{+} (ground state) of H 2 − superscript subscript H 2 {\rm H}_{2}^{-} . Lowest longitudinal vibrational energy presented as well as dissociation energies into two decay channels. Energies and distances in a.u.

$B\left({10}^{9}G\right)$	$E_T^{H_2^{-}}$	$R_{eq}^{H_2^{-}}$	$E_{vib}^0$	$E_{diss}^{H_2^{-}\to H^{-}+H}$	$E_{diss}^{H_2^{-}\to H_2+e}$
$235.0$	-8.02	0.352		0.05	-0.16
$500.0$	-11.18	0.300	0.11	-	0.49
$1000.0$	-14.29	0.247	0.16	-	0.74
$2350.0$	-18.92	0.188	0.26	2.79	0.95
$5000.0$	-23.85	0.151	0.37	-	-
$10000.0$	-29.26	0.128	0.52	-	-

Table 3. Table 3: H 3 subscript H 3 {\rm H}_{3} (excited states): Total energy E T subscript 𝐸 𝑇 E_{T} and equilibrium distance R e ​ q subscript 𝑅 𝑒 𝑞 R_{eq} (for R + = R − subscript 𝑅 subscript 𝑅 R_{+}=R_{-} ) for states ( − 4 ) + 4 {}^{4}(-4)^{+} and ( − 5 ) + 4 {}^{4}(-5)^{+} . Energies and distances in a.u.

$B\left({10}^{9}G\right)$	$E_T^{H_3\left[{}_{}{}^4(-4)_{}^{+}\right]}$	$R_{eq}^{H_3\left[{}_{}{}^4(-4)_{}^{+}\right]}$		$E_T^{H_3\left[{}_{}{}^4(-5)_{}^{+}\right]}$	$R_{eq}^{H_3\left[{}_{}{}^4(-5)_{}^{+}\right]}$
$100$	-8.15	0.50
$500$	-15.31	0.30
$1000$	-19.89	0.23		-19.37	0.24
$2350$	-26.82	0.18		-26.29	0.19

Table 4. Table 4: H 2 − superscript subscript H 2 {\rm H}_{2}^{-} (excited states): Total energy E T subscript 𝐸 𝑇 E_{T} and equilibrium distance R e ​ q subscript 𝑅 𝑒 𝑞 R_{eq} for states ( − 4 ) + 4 {}^{4}(-4)^{+} , ( − 5 ) + 4 {}^{4}(-5)^{+} . Energies and distances in a.u.

$B\left({10}^{9}G\right)$	$E_T^{H_2^{-}\left[{}_{}{}^4(-4)_{}^{+}\right]}$	$R_{eq}^{H_2^{-}\left[{}_{}{}^4(-4)_{}^{+}\right]}$		$E_T^{H_2^{-}\left[{}_{}{}^4(-5)_{}^{+}\right]}$	$R_{eq}^{H_2^{-}\left[{}_{}{}^4(-5)_{}^{+}\right]}$
$352.5$	-9.34	0.33
$500$	-10.72	0.30
$1000$	-13.72	0.26		-13.46	0.27
$2000$	-17.30	0.21		-17.06	0.22
$2350$	-18.22	0.20		-17.99	0.20

Equations

$$\mathcal{H}$$

$$+\frac{B}{2}(L_z+2S_z)+\frac{Z_{\mathtt{A}}{Z}_{\mathtt{B}}}{R_{-}}+\frac{Z_{\mathtt{A}}{Z}_{\mathtt{C}}}{R_{+}+R_{-}}+\frac{Z_{\mathtt{B}}{Z}_{\mathtt{C}}}{R_{+}},$$

$$\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)=e^{{\alpha }_{12}{r}_{12}+{\alpha }_{13}{r}_{13}+{\alpha }_{23}{r}_{23}}$$

$$\times \left(\prod _{k=1}^3{\rho }_k^{|m_k|}{e}^{i{m}_k{\varphi }_k}{e}^{-{\alpha }_{k\mathtt{A}}{r}_{k\mathtt{A}}-{\alpha }_{k\mathtt{B}}{r}_{k\mathtt{B}}-{\alpha }_{k\mathtt{C}}{r}_{k\mathtt{C}}-\frac{B}{4}{\beta }_k{\rho }_k^2}\right),$$

$$\Psi ({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)=(1+{\sigma }_N{\hat{P}}_{\mathtt{AC}})\hat{A}[\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)\chi ],$$

$$\hat{A}=1-{\hat{P}}_{12}-{\hat{P}}_{13}-{\hat{P}}_{23}+{\hat{P}}_{231}+{\hat{P}}_{312},$$

$$B_c^{{\mathrm{H}}_3}(0)\sim 1.9\times 10^{11}\text{mbox}G,$$

$$B_c^{{\mathrm{H}}_2^{-}}(0)\sim 2.7\times 10^{11}\text{mbox}G,$$

Full text

Surface magnetic fields $B\sim 10^{11}-10^{14}\,$G exist for many neutron stars, while for magnetized white dwarfs the magnetic field can reach $B\sim 10^{9}\,$G (see e.g. Science:2004 ; Garcia:2016 ). Among these stars there are many which have atmosphere, which is mostly composed from electrons and protons while some heavy nucleus (e.g. $\alpha,O,Ne,C,Fe$) can be also present. It is natural to assume that for surface temperatures of order of 10 eV, electrons and protons can condense into hydrogenic species, forming Hydrogen atoms and Hydrogenic molecular ions.

These enormous magnetic fields modify dramatically the structure of atoms and molecules: as the strength of the magnetic field increases the atoms and molecules become more bound and more compact, their electronic clouds get pronounced cigar-type form and eventually the charged centers are aligned on a magnetic line. Thus, the Coulomb systems become quasi-one-dimensional. Such strong magnetic fields eventually lead to the appearance of exotic charged molecular and atomic systems which do not exist without a strong magnetic field (see e.g. Turbiner:2006 and references therein). In particular, the pioneering studies by Ruderman Ruderman:1974 and, independently, by Kadomtsev-Kudryavtsev Kadomtsev:1970 ; Kadomtsev:1971a predicted qualitatively that finite and supposedly even infinite neutral hydrogenic (linear) chains (and Wigner crystals) could exist if magnetic fields are sufficiently strong. In particular, accurate calculations have shown that in addition to the hydrogen atom ${\rm H}$ and the ${\rm H}_{2}^{+}$ molecular ion, which exist for any magnetic field, at magnetic fields $B\gtrsim 10^{11}\,$G three protons situated along the magnetic line can be bound by single electron (!) forming the exotic molecular ion ${\rm H}_{3}^{2+}$ in linear configuration Turbiner:1999 . Furthermore, ${\rm H}_{3}^{2+}$ turns out to be the most bound one-electron hydrogenic system for magnetic fields $B\gtrsim 10^{13}\,$G Turbiner:2006 . This discovery was used later to construct a model of the atmosphere of the isolated neutron star 1E1207.4-5209 (see  Turbiner:2004m ) to explain the origin of the absorption lines at $\sim 0.7$ and $1.4\,$KeV detected by Chandra  Sanwal:2002 and confirmed by XMM-Newton Bignami:2003 X-ray observatories. It predicts the surface magnetic field of $(4\pm 2)\times 10^{14}$ G.

For systems with two or more electrons it is known that in sufficiently strong magnetic fields the ground state appears in the configuration where all spins of electrons are antiparallel to the magnetic field direction, hence, the total spin takes its maximal value. In field free case the total spin usually does not take the maximal value. It implies that the ground state type can evolve with magnetic field strength. This phenomenon was quantitatively observed for the first time for ${\rm H}_{2}$ molecule, where it was shown that the ground state changes from spin-singlet state ${}^{1}\Sigma_{g}$ at zero and weak magnetic fields to triplet unbound (repulsive) state ${}^{3}\Sigma_{u}$ for intermediate fields $B\gtrsim 5\times 10^{8}\,$G   while for larger magnetic fields $B\gtrsim 3\times 10^{10}\,$G the ground state is the spin-triplet state ${}^{3}\Pi_{u}$ Detmer:1997 (and references therein). It implies that in this domain ${\rm H}_{2}$ molecule is unstable towards dissociation ${\rm H}+{\rm H}$. As for the molecular ion ${\rm H}_{3}^{+}\,$in a strong magnetic field in linear parallel configuration (aligned with the magnetic field direction) the ground state of ${\rm H}_{3}^{+}\,$changes from spin-singlet ${}^{1}\Sigma_{g}$ state for weak magnetic fields $B\lesssim 5\times 10^{8}\,$G (at magnetic fields close to zero the ground state of ${\rm H}_{3}^{+}\,$is equilateral triangular configuration in spin-singlet state, it is the most bound hydrogenic specie) to a weakly-bound spin-triplet ${}^{3}\Sigma_{u}$ for intermediate fields and, eventually, to spin-triplet state ${}^{3}\Pi_{u}$ for magnetic fields $B\gtrsim 10^{10}\,$G, it is always stable Turbiner:2007 . The list of one-, two-electron hydrogenic systems, which can exist in a strong magnetic field being stable, is given in Turbiner:2006 ; Turbiner:2010 .

In general, it is known very little about atomic-molecular systems with three electrons in a strong magnetic field. In particular, the neutral hydrogenic chain ${\rm H}_{3}\,$was studied in  Lai-Salpeter:1992 along with finite hydrogenic chains ${\rm H}_{n}\,,\,\scriptstyle n=2,3,4\ldots$ in strong magnetic fields $B\geq 10^{11}$ G in sophisticated multiconfigurational Hartree-Fock method. The accuracy obtained was limited, grossly overestimated and in some cases the results were indicated as non-reliable (spurious), the question of the existence and stability was never discussed. We are not aware on any studies of negative hydrogenic molecular ion H${}_{2}^{-}$.

In present Letter the linear molecular chain ${\rm H}_{3}\,$  and the H${}_{2}^{-}$ ion are studied in a strong magnetic field being situated along a magnetic line (we call it the parallel configuration, see Fig. 1). Magnetic field is assumed strong enough to have minimal total spin projection $S_{z}=-3/2$, thus, we focus on states with total spin 3/2, and total magnetic quantum numbers $M=-2,-3,-4,-5$. We explore the magnetic fields $10^{11}\,\rm G\leq B\leq 10^{13}\,$G, where a non-relativistic approach with static nucleus is still valid, see for discussion Lai-Salpeter:1992 . The study is developed in the Born-Oppenheimer approximation of zero order, i.e. the nuclei are considered to be infinitely massive. The main goal of the study is to localize the domain of existence and stability of ${\rm H}_{3}\,$and H${}_{2}^{-}$ in strong magnetic fields. We consider two main dissociation channels for both ${\rm H}_{3}\,$: (i) ${\rm H}_{3}\rightarrow{\rm H}_{2}+{\rm H}$, (ii) ${\rm H}_{3}\rightarrow{\rm H}_{3}^{+}+{\rm e}$, and H${}_{2}^{-}$: (iii) ${\rm H}_{2}^{-}\rightarrow{\rm H}^{-}+{\rm H}$, (iv) ${\rm H}_{2}^{-}\rightarrow{\rm H}_{2}+{\rm e}$ . Dissociation energies are defined accordingly, $E_{diss}^{(i)}=E_{T}^{{\rm H}_{2}+{\rm H}}-E_{T}^{{\rm H}_{3}}\,,$ $E_{diss}^{(ii)}=E_{T}^{{\rm H}_{3}^{+}+e}-E_{T}^{{\rm H}_{3}}\,$,  and $E_{diss}^{(iii)}=E_{T}^{{\rm H}^{-}+{\rm H}}-E_{T}^{{\rm H}_{2}^{-}}\,,$ $E_{diss}^{(iv)}=E_{T}^{{\rm H}_{2}+e}-E_{T}^{{\rm H}_{2}^{-}}\,$. Note that writing about dissociation we ignore the conservation of total angular momentum projection due to possible presence of photon(s) in the final state and assume each of the final products is in the lowest energy state. The energy of a free electron in a magnetic field with $S_{z}=-1/2$ is zero (see He2pTurbinerVieyra for details). The lowest longitudinal vibrational states around the equilibrium configuration of the ground state are briefly studied.

Atomic units are used throughout ($\hbar=m_{e}=e=1$) in making calculations. For the magnetic field $B$ written in a.u. the conversion $1\,\mbox{a.u.}\,=2.35\times 10^{9}$ G is used, as for the energy $1\,a.u.=27.2$ eV .

Generalities. The Hamiltonian describing the Coulomb system of three electrons and three aligned (infinitely) massive charged centers $\tt A,B,C$ subject to a constant uniform magnetic field, parallel to the molecular axis ${\bf B}=B\,\hat{e}_{z}$, is given by

[TABLE]

where $\Delta_{i}$ is Laplacian for the $i$-th electron, $i=1,2,3$. $Z_{\tt A,B,C}=1$ are the charges of the heavy centers, $\eta=\tt A,B,C$ (protons). The term $-\frac{Z_{\eta}}{r_{i\eta}}$ corresponds to the Coulomb attractive potential between the electron $i$ and the nuclei $\eta$ with $r_{i\eta}$ being the electron-nuclei distance. The term $\frac{1}{r_{ij}}$ stands for the inter-electron repulsion between electrons $i$ and $j$, where $r_{ij}$ is the distance between electrons. In turn, $\frac{Z_{\tt A}Z_{\tt B}}{R_{+}}$, $\frac{Z_{\tt A}Z_{\tt C}}{R_{+}+R_{-}}$ and $\frac{Z_{\tt B}Z_{\tt C}}{R_{-}}$ are the classical Coulomb repulsion energy terms between the (fixed) $\tt A,B,C$ nuclei. The internuclear (classical) distances are $R_{+}$ and $R_{-}$ (see Fig.1). The Hamiltonian (Finite Hydrogenic molecular chain ${\rm H}_{3}\,$ and ion H${}_{2}^{-}$ exist in a strong magnetic field) includes the Zeeman terms $\frac{1}{2}{\bf B}\cdot{\bf L}=\frac{1}{2}B\,L_{z}$, and $\frac{g}{2}{\bf B}\cdot{\bf S}=B\,S_{z}$ (with the g-factor, $g=2$) and the diamagnetic terms $\frac{B^{2}}{8}\rho_{i}^{2}\,,$ where $\rho_{i}^{2}=x_{i}^{2}+y_{i}^{2}\,,(i=1,2,3)$ is the transverse distance (squared) between the $i$-th electron and the molecular axis. Putting $Z_{\tt B}=0$ the Hamiltonian (Finite Hydrogenic molecular chain ${\rm H}_{3}\,$ and ion H${}_{2}^{-}$ exist in a strong magnetic field) describes the 3-electron 2-center system in a magnetic field which may correspond to H${}_{2}^{-}$ molecule.

For the linear ${\rm H}_{3}$ molecule the equilibrium configuration was assumed symmetric: $R_{+}=R_{-}\equiv R$ – it is confirmed by calculations. In this configuration the Hamiltonian (Finite Hydrogenic molecular chain ${\rm H}_{3}\,$ and ion H${}_{2}^{-}$ exist in a strong magnetic field) is invariant under transformations $z_{1,2,3}\rightarrow-z_{1,2,3}$, and also invariant under rotation around the molecular axis. Thus, the conserved quantities we take into account are: the parity ($\pm$) w.r.t. to $z\rightarrow-z$, see Fig.1, the $z$-component of the total angular momentum $M_{z}$ (magnetic quantum number), the total electronic spin $S$ and its projection $S_{z}$ along the $z$-axis. These conserved quantities characterize the state of the system in the standard spectroscopic notation ${}^{2S+1}(M_{z})^{\pm}$.

The variational method is applied to find the energy of some low-lying states using a trial function based on physics relevance criteria. These trial functions allow us to reproduce both the Coulomb singularities and the correct asymptotic behavior of the potential at large distances (see, e.g. Turbiner:1984 ).

Trial Function. The orbital (spatial) part of the trial function is proposed as a product of screened lowest Coulomb orbitals, Landau orbitals and exponential correlation factors:

[TABLE]

where $\rho_{k}$, $\phi_{k}$ are the radial/angular cylindrical coordinates of the $k$-th electron with the magnetic quantum number $m_{k}$, $k=1,2,3$. In turn, the parameters $\alpha_{k\eta}$, $\alpha_{ij}$ with $i<j=1,2,3$, $\eta=\tt A,B,C$ are screened effective charges, $\beta_{1,2,3}$ measure screening of the magnetic field, the classical internuclear distances $R_{\pm}$ are taken as variational parameters. The total number of variational parameters in (2) is $17$ (13 as for ${\rm H}_{2}^{-}$).

For states of total spin $S=3/2$ with projection $S_{z}=-3/2$, the spin part of the trial function is ${\chi=\beta(1)\beta(2)\beta(3)}\ ,$ where ${\beta(k)}\,,\ k=1,2,3$ represents the $S=1/2$ spinor of the $k$-th electron with negative spin projection. Thus, the properly symmetrized total wave function is given by

[TABLE]

where $\hat{P}_{\tt AC}$ is the operator of permutation of the two end nucleus $\tt A$ and $\tt C$ (see Fig.1), and $\sigma_{N}=\pm 1$ is $z$-parity. The operator $\hat{A}$ is the three particle antisymmetrizer

[TABLE]

acting on the coordinates of the electrons. Here $\hat{P}_{ij}$ is the operator of permutation of the electrons $i\leftrightarrow j$, and $\hat{P}_{ijk}$ stands for the permutation of $(123)$ into $(ijk)$.

Numerics for the calculation of the variational energy is described in He2pTurbinerVieyra . Computations were performed in parallel using the cluster Karen (ICN-UNAM) with 120 Intel Xeon processors at $\sim 2.70\,$GHz.

Results. Variational calculations of the system $(3p,3e)$  and $(2p,3e)$ in parallel configuration, see Fig.1, for magnetic fields $10^{11}\,$G$\leq B\leq 10^{13}\,$G are carried out using the trial function (2) for spin-quartet states with total spin 3/2 and for total magnetic quantum numbers $M=-2,-3,-4,-5$. For $M=-2$ both systems are unbound: one electron always goes to infinity in $z$-direction. In all other cases we find bound states with a well pronounced minimum in the energy at finite nuclear distances at $R_{+}=R_{-}$ for ${\rm H}_{3}\,$and for H${}_{2}^{-}$ systems. It is seen that the total energy and the equilibrium internuclear distances of ${\rm H}_{3}\,$ and H${}_{2}^{-}$ decrease monotonously with the magnetic field increase, the systems become more bound and compact, see Table 1 and 2, respectively. Among the states we study the state with $M=-3$ when the magnetic quantum numbers are $m_{1}=0,m_{2}=-1,m_{3}=-2$ (being in agreement to the Pauli principle, it guarantees zero Pauli force) realizes the ground state of both ${\rm H}_{3}\,$and ${\rm H}_{2}^{-}$. Taking different combinations of $m_{i}$ keeping $M=-3$ in (2) does not improve the energies. States with $M=-4,-5$ always lie above the ground state ${}^{4}(-3)^{+}$. Note that for this state the equilibrium distance is smallest with respect to other states, see Table 1 and 2. Dissociation energies as well as lowest longitudinal vibrational energies are presented in Table 1 and Table 2. Systematically, the total energies for ${\rm H}_{3}\,$ are smaller than ones calculated in Lai-Salpeter:1992 , while equilibrium distances are comparable.

For $B\gtrsim 10^{11}$ G the molecule ${\rm H}_{3}\,$is always stable towards the dissociation channel (i) ${\rm H}_{3}^{+}+e$, see Table 1. However, as for the channel (ii) ${\rm H}_{3}\rightarrow{\rm H}_{2}+{\rm H}$ there is a critical magnetic field, where dissociation energy for the ground state vanishes,

[TABLE]

It indicates that the linear molecular chain ${\rm H}_{3}\,$always exists, it becomes stable at $B>B_{c}^{{\rm H}_{3}}(0)$ being metastable for smaller magnetic fields. The stability of the linear molecule ${\rm H}_{3}\,$ was checked towards longitudinal symmetric vibrations (s), $R_{+}=R_{-}$ and antisymmetric vibrations (a) $R_{+}\neq R_{-}$. The lowest vibrational energy of the symmetric mode $E^{(s)}_{vib}$ are always smaller than those of the antisymmetric mode $E^{(a)}_{vib}$. Both energies increase with the magnetic field growth, see Table 1. For $B\gtrsim 5\times 10^{11}\,$G the dissociation energies are always larger than the sum of the lowest vibrational energies, hence, the potential energy surface contains, at least, one vibrational state.

As for ${\rm H}_{2}^{-}$ ion the total energies, equilibrium distances and dissociation energies as well as lowest longitudinal vibrational energy are presented in Table 2. For $B\sim 10^{11}$ G ${\rm H}_{2}^{-}$ is unstable towards both channels (iii) and (iv). However, at the magnetic field $B\sim 2.35\times 10^{11}\,$G the channel (iii) gets forbidden while the channel (iv) is still open. With magnetic field increase at

[TABLE]

this channel gets also forbidden and H${}_{2}^{-}$ becomes stable, being metastable for smaller magnetic fields. The lowest vibrational energy increases with the magnetic field growth. Potential energy surface contains, at least, one vibrational state in all range of magnetic fields studied.

Excited States.  The excited states ${}^{4}(-4)^{+}$, ${}^{4}(-5)^{+}$ were studied for both ${\rm H}_{3}\,$and H${}_{2}^{-}$. For $B\gtrsim 10^{11}\,$G both states are bound for both systems. It was checked that minimal energy for ${\rm H}_{3}\,$ always corresponds to symmetric configuration $R_{+}=R_{-}$. The total energy and the equilibrium internuclear distance $R_{eq}$ for ${\rm H}_{3}\,$and H${}_{2}^{-}$ are presented in Table 3 and  4, respectively. For both states ${}^{4}(-4)^{+}$, ${}^{4}(-5)^{+}$ for each system there exist critical magnetic fields, see below section Conclusions, for which they become stable with respect to dissociation ${\rm H}_{3}\rightarrow{\rm H}_{2}+{\rm H}$ and ${\rm H}_{2}^{-}\rightarrow{\rm H}_{2}+e$, correspondingly.

*Conclusions. * We have shown that trihydrogen molecule ${\rm H}_{3}\,$in the form of linear chain exists for $B\gtrsim 10^{11}\,$G with the ground state ${}^{4}(-3)^{+}$. It becomes stable towards dissociation for magnetic fields larger than the critical magnetic field $B_{c}^{\rm H_{3}}(0)\simeq 1.9\times 10^{11}\,$G , it also remains stable towards small longitudinal vibrations. Its first excited state ${}^{4}(-4)^{+}$ is always bound and appears stable towards dissociation at $B_{c}^{\rm H_{3}}(1)\sim 6.9\times 10^{11}\,$G, while the second excited state ${}^{4}(-5)^{+}$ is stable toward dissociation for $B\gtrsim B_{c}^{\rm H_{3}}(2)\sim 1.2\times 10^{12}\,$G. It excludes a qualitative prediction Ruderman:1974 about existence of infinite hydrogenic chain (and Wigner crystal) for magnetic fields $B<B_{c}^{\rm H_{3}}(0)$ and, in fact, $\lesssim 10^{12}$ G. For any available strong magnetic field the absorption due dissociation of ${\rm H}_{3}\,$occurs at less than 100 eV, hence, could not be detected by Chandra or XMM X-ray observatories. As for linear molecular chain H4 one can estimate following the results of Turbiner:2010 for H2, Table 1 for H3 and Lai-Salpeter:1992 for H4 that it becomes stable for $B\gtrsim 10^{12}$ G. Dihydrogen negative molecular ion H${}_{2}^{-}$ with the ground state ${}^{4}(-3)^{+}$ is always bound and becomes stable towards dissociation for $B_{c}^{\rm H_{2}^{-}}(0)\simeq 2.7\times 10^{11}\,$G. Its first excited state ${}^{4}(-4)^{+}$ as well as the second excited state ${}^{4}(-5)^{+}$ are metastable towards dissociation, both states become stable at $B_{c}^{\rm H_{2}^{-}}(1)\sim 9.0\times 10^{11}\,$G and $B\gtrsim B_{c}^{\rm H_{2}^{-}}(2)\sim 1.9\times 10^{12}\,$G, respectively. Both molecular systems can be present in the magnetized neutron star atmosphere at large surface magnetic field but with not very hot surface temperature.

*Acknowledgements. * The authors thank CONACyT A1-S-17364 and DGAPA IN108815 grants (Mexico) for partial support.