The He${}_{2}^{+}$ molecular ion and the He^- atomic ion in strong magnetic fields

It seems obvious that the chemical composition of the atmosphere of a magnetic white dwarf or a magnetized neutron star can not be established as long as we lack reliable information about the behavior of many-body Coulomb systems, especially about simple molecules and atoms in the presence of strong magnetic fields. So far, only Hydrogen-like and Helium-like atomic systems, and H${}_{2}^{+}$-type molecular systems have been studied to a certain depth. There are indications that in addition to traditional atoms and molecules other non-standard, exotic atomic and molecular systems can also exist in strong magnetic fields (see for example, Turbiner and López Vieyra (2006); Turbiner et al. (2010)). In a recent discovery Turbiner and López Vieyra (2013) it was found that even the ${\rm He}^{-}$ atomic ion becomes stable for magnetic fields $B\gtrsim 0.13\,$a.u. with the spin doublet state ${}^{2}(-1)^{+}$ as ground state at first and then, for $B\gtrsim 0.74\,$a.u. all electron spins get aligned and antiparallel to a magnetic field direction: the corresponding state ${}^{4}(-3)^{+}$ becomes the ground state of the system (see also Salas and Varga (2014)). It happens in spite of the fact that Helium belongs to the most inert (closed shell) atomic systems. Needless to say that Helium has a rich chemistry even in the absence of intense magnetic fields (see e.g. Grandinetti (2004)).

Usually, investigations of the Coulomb systems in strong magnetic fields (unreachable in the lab) are justified by the fact there exists a strong magnetic field on the surface of many neutron stars $B\sim 10^{11-13}\,$G and of some highly-magnetized white dwarfs $B\sim 10^{8-9}\,$G, see e.g. review García-Berro et al. (2016). In general, magnetic fields can reach $B\sim 10^{15}\,$G, or even higher, in the case of the so-called magnetars - the neutron stars with anomalously large surface magnetic field. While there is the evidence for the presence of Helium in the atmosphere of magnetic white dwarfs Jordan et al. (1998), there is no similar understanding about Helium in whatsoever form in atmospheres of neutron stars, and in general about their chemical content that can satisfactorily explain the observations.

The discovery of absorption features at $\sim$0.7 KeV and $\sim$1.4 KeV in the X-ray spectrum of the isolated neutron star 1E1207.4-5209 by Chandra X-ray observatory Sanwal et al. (2002), and its further confirmation by XMM-Newton X-ray observatory Mereghetti et al. (2002) motivated to perform studies of atoms and molecules in a strong magnetic field. At present there is a number of neutron stars whose atmospheres are characterized by absorption features: all of them are waiting to be solidly explained. These observations make clear that a detailed study of traditional atomic-molecular systems is needed, as well as for a search for new exotic chemical compounds which exist in a strong magnetic field only (see e.g. Turbiner (2007); López Vieyra et al. (2007a); Turbiner et al. (2007); López Vieyra et al. (2007b)).

As a result of such investigations, a model of helium-hydrogenic molecular atmosphere of the neutron star 1E1207.4-5209 was proposed which is based on the assumption that the most abundant components in the atmosphere are the exotic molecular ions ${\rm He}_{2}^{3+}$ and ${\rm H}_{3}^{2+}$, with the presence of ${\rm He}^{+},({\rm HeH})^{2+},{\rm H}_{2}^{+}$ subject to a surface magnetic field $\approx 4.4\times 10^{13}$ G (see Turbiner (2005)). Conjectures about the absence of hydrogen envelopes in some neutron stars have also motivated the study of atmospheres composed of neutral helium. However, those simple models appear to be in conflict with observations. Models of atmosphere, composed with a large abundance of molecular systems containing helium, have been later suggested (see Turbiner (2005), van Kerkwijk and Kaplan (2007) and references therein). For reasons which are not completely clear to the present authors it has been emphasized van Kerkwijk and Kaplan (2007) that the ${\rm He}_{2}^{+}$ molecule has to play a particularly important role. This molecular system exists in a field-free case in the spin-doublet, nuclei-permutation-antisymmetric $(u)$, reflection-symmetric (with respect to any plane containing the internuclear axis) $(+)$ ground state ${}^{2}{}0^{+}_{u}$ Pauling (1933), usually denoted as ${}^{2}{\Sigma_{u}}^{+}$. It is a rather compact system characterized by a small dissociation energy $\sim 2.5$  eV into ${\rm He}^{+}({}^{2}{\rm S})+{\rm He}({}^{1}{\rm S})$. The lowest spin-doublet, nuclei-permutation-symmetric $(g)$ excited state ${}^{2}0^{+}_{g}$ Pauling (1933), usually denoted as ${}^{2}{\Sigma_{g}}^{+}$, is repulsive. It is essentially unbound with shallow van-der-Waals minimum at large internuclear distance, see Grandinetti (2004) and references therein. It took us a number of years to perform a quantitative study of this particular system in the presence of a strong magnetic field, which is the subject of the present work. We are not aware of any similar previous study.

It is quite common in the field-free case that the ground state of simple atoms and molecules be characterized by the lowest possible total electron spin. Since the first qualitative studies of atomic and molecular systems Kadomtsev (1970); Kadomtsev and Kudryavtsev (1971a, b, 1972); Ruderman (1971, 1974) it became clear that in sufficiently strong magnetic fields the ground state is eventually realized by a state where the spins of all electrons are antiparallel to the magnetic field direction. Thus, the total electron spin takes maximal value as well as its total projection. It implies that the ground state depends on the magnetic field strength: there exists one (or several) threshold magnetic field for which one type of ground state changes to another one. This phenomenon was quantitatively observed for the first time for the ${\rm H}_{2}$ molecule. It was shown that spin-singlet ground state ${}^{1}\Sigma_{g}$, for small magnetic field, changes for intermediate fields $B\gtrsim 0.2\,$a.u. to the unbound (repulsive) ${}^{3}\Sigma_{u}$ state as the ground state (precisely in the domain of magnetic fields typical of magnetic white dwarfs). While for stronger magnetic fields $B\gtrsim 12.3\,$a.u., the ground state of the hydrogen molecule is realized by spin-triplet state ${}^{3}\Pi_{u}$ (see Detmer et al. (1998) and references therein). Another recent example, which was mentioned above, is the case of the ${\rm He}^{-}$ atomic ion where the ground state is realized first by the spin doublet state ${}^{2}(-1)^{+}$ for $B\gtrsim 0.13\,$a.u., and later by the spin-quartet state ${}^{4}(-3)^{+}$, for magnetic fields $B\gtrsim 0.74\,$ a.u., where all electron spins are aligned antiparallel to a magnetic field direction. In general, the phenomenon of change of the ground state nature with a magnetic field strength in traditional atomic systems was known since a time ago, see e.g. Ivanov and Schmelcher (2001) and reference therein.

The aim of this article is to perform a variational study of the ${\rm He}_{2}^{+}$ molecular ion subject to a strong magnetic field in the state ${}^{4}(-3)^{+}_{g}$ , when all electron spins get oriented anti-parallel to the magnetic field direction, the electronic total angular momentum projection is equal to $M=-3$ ¹¹1To avoid a contradiction with the Pauli principle, thus, the appearance of the Pauli forces, it is further assumed that all three electrons have different magnetic quantum numbers, i.e. $m_{1}=0,m_{2}=-1,m_{3}=-2$, and to show that the system is stable towards all possible decays or dissociation. Further it is naturally assumed that ${}^{4}(-3)^{+}_{g}$ is the ground state. Due to extreme technical complexity we do not discuss other states of ${\rm He}_{2}^{+}$ and leave the question about evolution of the type of the ground state with magnetic field changes for a future publication. Since our study is limited to the question of the existence and stability of this system in a certain state, the main attention is devoted to the exploration of all possible decay channels. A natural assumption about the optimal (equilibrium) configuration with minimal total energy is one which is achieved in the parallel configuration: where the internuclear axis connecting the two massive $\alpha$-particles (${\rm He}$ nuclei) is situated along the magnetic line.

Another aim of the article is to continue to study the ${\rm He}^{-}$ atomic ion in spin-quartet state ${}^{4}(-3)^{+}$ for strong magnetic fields $B\leq 1$ a.u., which was initiated in Turbiner and López Vieyra (2013). This study is necessary due to the possible decay mode ${\rm He}_{2}^{+}$ into ${\rm He}^{-}+\alpha$.

The consideration is non-relativistic, based on a variational solution of the Schrödinger equation. The magnetic field strength is restricted to magnetic fields $B\leq 10000$ a.u. ($=2.35\times 10^{13}\,$G) below the relativistic Schwinger limit. Also it is based on the Born-Oppenheimer approximation of zero-th order: the particles of positive charge ($\alpha$-particles) are assumed to be infinitely massive.

Atomic units are used throughout ($\hbar=m_{e}=e=1$). The magnetic field $B$ is given in a.u., with $B_{0}=2.35\times 10^{9}$ G. For energies given in eV, the conversion $1\,{\rm a.u.}=27.2\,{\rm eV}$ was used. All energies, which are mentioned in the article, are the total energies (with spin terms included) if it is not indicated otherwise.

The non-relativistic Hamiltonian for a three-electron diatomic molecule with fixed nuclei $\sf A,B$ in uniform constant magnetic field $\bm{B}=B\bm{e}_{z}$, directed along $z$-axis, with vector potential in the symmetric gauge ${\bf A}=\frac{1}{2}{\mathbf{B}}\times{\mathbf{r}}$ is given by

$${\cal H}\ =-\sum_{i=1}^{3}\left(\frac{1}{2}{\nabla}_{i}^{2}\ +\ \sum_{\eta=\sf A,B}\frac{Z_{\eta}}{r_{i\eta}}\right)\ +\ \sum_{{i=1}}^{3}\sum_{{j>i}}^{3}\ \frac{1}{r_{ij}}\ +\frac{B^{2}}{8}\sum_{i=1}^{3}\ {\rho_{i}}^{2}+\frac{B}{2}(\hat{L}_{z}+2\hat{S}_{z})+\frac{Z_{\sf A}Z_{\sf B}}{R}\,,$$

(1)

where ${\nabla}_{i}$ is the 3-vector of the momentum of the $i$th electron, $Z_{\eta}$ is the charge of the nucleus $\eta=\sf A,B$, the terms $-Z_{\eta}/r_{i\,\eta},$ correspond to the Coulomb interactions of the electrons with each charged nuclei ($r_{i\,\eta}=|\bm{r}_{i\,\eta}|$ is the distance between the $i$-th electron and the $\eta$-nuclei), the three terms $1/r_{ij}$ ($j>i=1\ldots 3$) are the inter-electron Coulomb repulsive interactions ($r_{i\,j}$ are the distances between the $i,j$-th electrons), and the term $+Z_{\sf A}Z_{\sf B}/R$ is the classical Coulomb repulsion energy between the nuclei, where $R$ is the internuclear distance (see Fig. (1) for notations). The Hamiltonian (1) includes, the paramagnetic terms $\frac{1}{2}\ \bm{B}\cdot\bm{l_{i}}$ as well as the spin Zeeman-term $\bm{B}\cdot\bm{s}_{i}$ for the interaction of the magnetic field with the spin, and the diamagnetic term $\frac{\bm{B}^{2}}{8}{\rho_{i}}^{2}$, with ${\rho_{i}}^{2}=x_{i}^{2}+y_{i}^{2}$ for each electron, $i=1,2,3$. If the magnetic field is directed along the $z$ direction and parallel to the internuclear axis, the component of the total angular momentum along the $z$-axis $M$, the total spin $S$, the $z$ projection of the total spin $S_{z}$, and the total $z$ parity $\Pi_{z}$ are conserved quantities. Sometimes, during the text the spectroscopic notation $\nu^{2S+1}M^{\Pi_{z}}$ (with standard labels $\Sigma,\Pi,\Delta\ldots$ for $|M|=0,1,2$ etc) is used for the electronic states. Here $\nu$ stands for the degree of excitation for given (fixed) symmetry. In our case of a homonuclear diatomic molecule an additional subscript $g/u$ (gerade/ungerade) indicates a symmetric/antisymmetric state with respect to the permutation of the identical nuclei.

Before approaching to concrete calculations a description of the ground state of a Coulomb system of $k$ electrons and several heavy charged centers in a strong magnetic field should be given. In fact, a complete qualitative picture was presented in the pioneering works by Kadomtsev-Kudryavtsev  Kadomtsev (1970); Kadomtsev and Kudryavtsev (1971b, a, 1972) and Ruderman Ruderman (1971, 1974):

Observation.

(i) All spins of electrons are oriented antiparallel to the magnetic field direction. Hence, the total electronic spin projection is $-\frac{k}{2}$ ,

(ii) All heavy centers are situated on a magnetic line. Hence, there is no gyration,

(iii) Electronic magnetic quantum numbers are different and take values $0,-1,-2,\ldots,~{-(k-1)}$, hence, the total magnetic quantum number of the system $M=-\frac{k(k-1)}{2}$. This configuration does not contradict to the Pauli principle and implies vanishing Pauli forces.

We are not familiar with a rigorous proof of the validity of this observation in general.

For Hydrogen atom, $k=1$, validity of this observation, see item (iii), was explicitly checked by Ruder et al Ruder et al. (1994) and for other one-electron systems in Turbiner and López Vieyra (2006) - it was shown the ground state corresponds to $M=0$. For Helium atom it was checked in Becken and Schmelcher (2000) - the ground state was $(1s_{0}2p_{-1})$ type, thus, $M=-1$ and for Lithium atom it was $(1s_{0}2p_{-1}3d_{-2})$ type, thus, $M=-3$ Al-Hujaj and Schmelcher (2004) as well as for ${\rm He}^{-}$ Turbiner and López Vieyra (2013). For two-electron molecules ${\rm H}_{2},{\rm HeH}^{+},{\rm He}_{2}^{++}$ it was checked that lowest energy occurs at $M=-1$ comparing to $M=0,-2$, see Detmer et al. (1998),Turbiner and Guevara (2007),Turbiner et al. (2010), respectively.

The variational method is used to study the state ${}^{4}(-3)^{+}_{g}$ of the ${\rm He}_{2}^{+}$ molecular ion (with infinitely massive centers) placed in a uniform magnetic field parallel to the molecular $z$-axis. In general, the wavefunction of the electronic Hamiltonian (1) with two identical nuclei can be written in the form

$$\psi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})=(1+\sigma_{N}\,P_{\sf AB}){\cal A}\left[\,\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})\chi\,\right]\,,$$

(2)

where $\chi$ is a three-electron spin eigenfunction corresponding either to a total electronic spin $S=1/2$ or $S=3/2$. Here ${\mathbf{r}}_{1,2,3}$ are position vectors of the first, second, third electrons, respectively, see Fig.1. The function $\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})$ is a three particle orbital function, $P_{\sf AB}$ is the permutation operator of the two identical nuclei ($\sigma_{N}=\pm 1$ for gerade/ungerade states, respectively). The operator ${\cal A}$ is the three-particle antisymmetrizer:

$${\cal A}=1-P_{12}-P_{13}-P_{23}+P_{231}+P_{312}\,.$$

(3)

Here, $P_{ij}$ represents the permutation $i\leftrightarrow j$, and $P_{ijk}$ stands for the permutation of $123$ into $ijk$. For strong magnetic fields which are typically present in the atmosphere of neutron stars, a natural expectation is that the ground state corresponds to the case when all electron spins are aligned antiparallel to the magnetic field, i.e. $S_{z}=-3/2$ and, thus, with the three-electron spin eigenfunction $\chi$ being totally symmetric. In this case, when $S_{z}=-3/2$, the trial function is written as

$$\psi^{S_{z}=-3/2}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})=(1+\sigma_{N}\,P_{\sf AB}){\cal A}\left[\,\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})\,\right]\beta(1)\beta(2)\beta(3)\,,$$

(4)

with $\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})$ a properly chosen orbital function, which then antisymmetrized by ${\cal A}$. Here $\beta(i),\ i=1,2,3$ is spin down function of $i$th electron (see below).

On the other hand, for states of the total spin projection $S_{z}=-1/2$ ($S=1/2$) we have two linearly independent spin eigenfunctions:

	$\displaystyle\chi_{1}$	$\displaystyle=\frac{1}{\sqrt{2}}[\beta(1)\alpha(2)\beta(3)-\alpha(1)\beta(2)\beta(3)]\ ,$		(5)
	$\displaystyle\chi_{2}$	$\displaystyle=\frac{1}{\sqrt{6}}[2\beta(1)\beta(2)\alpha(3)-\beta(1)\alpha(2)\beta(3)-\alpha(1)\beta(2)\beta(3)]\ ,$		(6)

where $\alpha(i)(\beta(i)),\ i=1,2,3$ are spin up (spin down) functions of the $i$th electron. So, the general form of a spin projection $S_{z}=-1/2$ function has the form

$$\Phi=\phi_{1}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})\chi_{1}+\phi_{2}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})\chi_{2}\,,$$

(7)

where $\phi_{1}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})$ and $\phi_{2}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})$ are orbital functions. In particular we can choose these functions to be proportional and write

$$\Phi=\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})(\chi_{1}+a\chi_{2})\,,$$

(8)

where $a$ is a variational parameter²²2A similar treatment was used for the study of the Li atom in a magnetic field in Guevara et al. (2009).. Thus, the trial function for states of spin $S_{z}=-1/2$ can be written as (cf. (2))

$$\psi^{S_{z}=-1/2}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})=(1+\sigma_{N}\,P_{\sf AB}){\cal A}\left[\,\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})(\chi_{1}+a\chi_{2})\,\right]\,,$$

(9)

with $\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})$ a properly chosen orbital function.

Variational trial functions are designed following physical relevance arguments. In particular, we construct wavefunction which allow us to reproduce both the Coulomb singularities of the potential and the correct asymptotic behavior at large distances (see, e.g. Turbiner (1984)). Following such criterion we propose the function

$$\phi({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{3})=\prod_{k=1}^{3}\left(\rho_{k}^{|m_{k}|}e^{i{m_{k}}\phi_{k}}e^{-\alpha_{k,{\sf A}}r_{k{\sf A}}-\alpha_{k,{\sf B}}r_{k{\sf B}}}e^{-\frac{B}{4}\beta_{k}\rho_{k}^{2}}\right)e^{\alpha_{12}r_{12}+\alpha_{13}r_{13}+\alpha_{23}r_{23}}\ ,$$

(10)

which is a type of product of Guillemin-Zener type molecular orbital functions multiplied by the product of Landau type orbitals for an electron in a magnetic field. Here $\alpha_{k_{\sf A}},\alpha_{k_{\sf B}}$ and $\beta_{k}$, $k=1,2,3$ are parameters. In the case of the fully polarized state $S=3/2$ and in order to avoid a contradiction with the Pauli principle, it is further assumed that all electrons have different magnetic quantum numbers in a certain minimal way: $m_{1}=0,m_{2}=-1,m_{3}=-2$, hence, the total electronic quantum number is $M=-3$. It was already discussed in Turbiner et al. (2010) that this assumption seems obviously correct in the case of atoms and atomic ions, where the electrons are sufficiently close to each other, but not that obvious for the case of molecules for which the electrons are spread in space. All of them (or, at least, some of them) can be in the same quantum state, with the same spin projection and magnetic quantum number. This situation was observed for ${\rm H}_{2}$ and ${\rm H}_{3}^{+}$, where in a domain of large magnetic fields the ground state was given by the state of maximal total spin but with the electrons having the same zero magnetic quantum number (see Turbiner et al. (2010) and references therein). However, for very strong fields the state of minimal total energy always corresponds to the situation described above with all electrons having different magnetic quantum numbers.

In (10) the variational parameters $\alpha_{k,{\sf A}},\alpha_{k,{\sf B}}$ ($k=1,2,3$) have the meaning of screening (or anti-screening) factors (charges) for the nucleus ${\sf A,B}$ respectively, as it is seen from the $k$-th electron. The variational parameters $\beta_{k}$ are screening (or anti-screening) factors for the magnetic field seen from $k$-th electron, and the parameters $\alpha_{ij},i<j=1\ldots 3$ “measure” the screening (or anti-screening) of the inter-electron interaction. In total, the trial function (10) has 12 variational parameters in addition to the internuclear distance $R$ which can also be considered as a variational parameter.

The calculation of the variational energy using the trial function (4)-(10) involves two major parts: (i) 9-dimensional numerical integrations which were implemented by an adaptive multidimensional integration routine (cubature)Genz and Malik (1980), and (ii) a minimizer which was implemented with the minimization package TMinuit from CERN-lib (an old version of TMinuitMinimizer in the ROOT systemBrun and Rademakers (1997), which allows fixing/releasing parameters, was recovered and adapted to our purposes). The 9-dimensional integrations were carried out using a dynamical partitioning procedure: the domain of integration is manually divided into sub-domains following the profile of the integrand. Then each sub-domain is integrated using the routine CUBATURE. In total, we have a subdivision to $\sim 2000$ subregions for the numerator and $\sim 2000$ for the denominator in the variational energy. With a maximal number of sampling points $\sim 2\times 10^{8}$ for the numerical integrations (it guarantees the relative accuracy $\sim 10^{-3}-10^{-4}$ in integration) for each subregion, the time needed for one evaluation of the variational energy takes $5\times 10^{4}\,$ seconds with 96 processors. It was checked that this procedure stabilizes the estimated accuracy to be reliable in the first two decimal digits. However, in order to localize a domain, where minimal parameters are, the minimization procedure with much less number of sample points was used in each sub-domain and a single evaluation of the energy usually took $\sim 15-20\,$mins. Once a domain is roughly localized the number of sample points increased by factor $\sim 10^{2}$. Final evaluation was made with $2\times 10^{8}$ sampling points and for the strongest fields $B=100,1000,10000$ a.u. it was even $5\times 10^{8}$ with a subdivision of 7 subintervals in each $z$-domain. Typically, a minimization procedure required several hundreds of evaluations. Computations were performed in parallel with a cluster Karen with 96 Intel Xeon processors at $\sim 2.70$GHz.

In this section we analyze different decay channels of the $S_{z}=-3/2$, $M=-3$ state ${}^{4}(-3)^{+}_{g}$ of the ${\rm He}_{2}^{+}$ molecular ion in a magnetic field in the range $1\,{\rm a.u.}\leq B\leq 10000\,$a.u. Possible decay channels that we consider for the system $(\alpha\alpha eee)$ placed in a magnetic field are

$$\begin{array}[]{llr}{\rm He}_{2}^{+}&\to{\rm He}+{\rm He}^{+}&(a)\\ &\to{\rm He}^{-}+{\alpha}&(b)\\ &\to{\rm He}+\alpha+e&(c)\\ &\to{\rm He}^{+}+{\rm He}^{+}+e&(d)\\ &\to{\rm He}_{2}^{2+}+e&(e)\\ &\to{\rm He}_{2}^{3+}+e+e&(f)\\ \end{array}$$

A few remarks emerge immediately: in the cases where there are free electrons ³³3For the total energy of a free electron $E_{e}$ (excluding the spin contribution) in the magnetic field in the symmetric gauge the $z$-component of the angular momentum $L_{z}$ is conserved and the electron Landau levels are $E_{e}=\hbar\omega_{B}(n+1/2)$, where $n=n_{\rho}+\frac{|m|+m}{2}=0,1,2\ldots$ All $m\leq 0$ states are degenerate. Here $\omega_{B}=\frac{eB}{m_{e}c}$ is the cyclotron frequency. in the decay channel (channels (c)-(f)) we can assume that the electrons are in the ground state $n=0,s_{z}=-1/2,m<0$ which yields $E=0$ (regardless of the magnetic quantum number $m$ carried by the electron, where the spin contribution is included). In Born-Oppenheimer approximation the energy of a free $\alpha$-particle in a magnetic field is zero by assumption. Also, it is known that the helium atom exists for any magnetic field strength. For magnetic fields $0\leq B\lesssim 0.75\,$a.u. the spin-singlet state $1{}^{1}{0}^{+}$ is the ground state. For $B\gtrsim 0.75\,$ a.u., the spin-triplet state (with $m=-1$) $1{}^{3}(-1)^{+}$ becomes the ground state. For the ${\rm He}$ atom in a magnetic field $B\leq 100$ a.u., the corresponding total energies collected in Table (3) were taken from references Becken et al. (1999); Becken and Schmelcher (2000). Such energies were calculated for the infinite nuclear mass approximation and include the spin contributions $B\,s_{z}$ for each electron. For magnetic fields $B=1000$ and $10000$ a.u. a simple variational Ansatz with 5 variational parameters was used to estimate the value of the energy of the spin-triplet state of Helium (see section VIII below for more details).

To obtain the energy of the ${\rm He}^{+}$ ion we use the basic result of Surmelian and O’Connel for hydrogen-like atoms

$$E(Z,B)=Z^{2}E(1,B/Z^{2})\,,$$

(11)

and use the data of Kravchenko et al. (1996) for the binding energies of Hydrogen in a magnetic field (recalling that instead of the total energy $E_{T}$, in Kravchenko et al. (1996) the authors reported the energies $E_{b}=(1+m+|m|)\gamma/2-E$, which coincide with the binding energies $\epsilon=\gamma/2-E$ for $m<0$). The Zeeman contribution to the total energy $B\,s_{z}$ due to the spin of the electron is not taken into account in the results appearing in Kravchenko et al. (1996). Such contribution for the case of a spin antiparallel to the magnetic field ($s_{z}=-\frac{1}{2}$) is $E_{\rm spin}=-B/2\,$a.u. and was added to the results collected in Table 3 to obtain the total energies of ${\rm He}^{+}$.

The molecular system $(\alpha\alpha e)$ (${\rm He}_{2}^{3+}$ molecular ion): accurate variational calculations in equilibrium configuration parallel to the magnetic field for the ground state $1\sigma_{g}$ were carried out in detail in Turbiner and López Vieyra (2006, 2007) for the range of magnetic fields $100\,{\rm a.u.}\lesssim B\lesssim B_{\rm Schwinger}$ where $B_{\rm Schwinger}=4.414\times 10^{13}\,$G is the non-relativistic limit. It was found that for magnetic fields $10^{2}\lesssim B\lesssim 10^{3}\,$a.u. the system ${\rm He}_{2}^{3+}$ is unstable towards decay to ${\rm He}^{+}+\alpha$. Thus, in principle we can neglect in our considerations the decay channel (e) above. Nonetheless, at $B\gtrsim 10^{4}\,$ a.u., this compound becomes the system with the lowest total energy among the one-electron helium (helium-hydrogen) chains (for details see Turbiner and López Vieyra (2006)).

The molecular system $(\alpha\alpha ee)$ ${\rm He}_{2}^{2+}$ molecular ion: this molecule was studied in detail in Turbiner and Guevara (2006) in the domain of magnetic fields $B=0-B_{\rm Schwinger}$. It was shown that the lowest total energy state depends on the magnetic field strength and evolves from the spin-singlet ${}^{1}\Sigma_{g}$ metastable state at $0\leq B\lesssim 0.85\,$ a.u. to a repulsive spin triplet ${}^{3}\Sigma_{u}$ (unbound state) for $0.85\lesssim B\lesssim 1100\,$a.u. and then to a strongly bound triplet state ${}^{3}\Pi_{u}$ state. Hence, there exists quite a large domain of magnetic fields where the ${\rm He}_{2}^{2+}$ molecule is unbound and represented by two atomic helium ions in the same electron spin state but situated at an infinite distance from each other.

In order to have a complete understanding about the stability of the molecular ${\rm He}_{2}^{+}$ ion in magnetic fields, we need to extend the study on the ${\rm He}^{-}$ atomic ion (three electron atomic system $(\alpha,e,e,e)$) in magnetic fields to the regime of very strong fields $B\gg 1$a.u. In this section we review the basic notions for the study of the ${\rm He}^{-}$ ion in magnetic fields. In Turbiner and López Vieyra (2013) it was found that the ground state of ${\rm He}^{-}$ in a magnetic field is realized by a spin-doublet ${}^{2}(-1)^{+}$ at $0.74\,{\rm a.u.}\gtrsim B\gtrsim 0.13\,{\rm a.u.}$ and it becomes a fully polarized spin-quartet ${}^{4}(-3)^{+}$ for larger magnetic fields. Thus, we will extend that study of the ${\rm He}^{-}$ ion in strong magnetic fields, in the fully polarized, spin quartet $S=3/2$, state only. For more details the reader is addressed to the reference Turbiner and López Vieyra (2013).

The non-relativistic Hamiltonian for an atomic system of three-electron and one infinitely massive center of charge $Z$ in a magnetic field (directed along the $z$-axis and taken in the symmetric gauge) is

$$\begin{split}{\cal H}\ &=-\sum_{k=1}^{3}\left(\frac{1}{2}{\nabla}_{k}^{2}\ +\ \frac{Z}{r_{k}}\right)\ +\ \sum_{{k=1}}^{3}\sum_{{j>k}}^{3}\ \frac{1}{r_{kj}}\ +\frac{B^{2}}{8}\sum_{k=1}^{3}\ {\rho_{k}}^{2}+\frac{B}{2}(\hat{L}_{z}+2\hat{S}_{z})\,,\end{split}$$

(12)

where ${\nabla}_{k}$ is the 3-vector momentum of the $k$th electron, $r_{k}$ is the distance between the $k$th electron and the nucleus, $\rho_{k}$ is the distance of the $k$th electron to the $z$-axis, and $r_{kj}$  $(k,j=1,2,3)$ are the inter-electron distances. $\hat{L}_{z}$ and $\hat{S}_{z}$ are the $z$-components of the total angular momentum and total spin operators, respectively. Both $\hat{L}_{z}$ and $\hat{S}_{z}$ are integrals of motion and can be replaced in (1) by their eigenvalues $M$ and $S_{z}$ respectively. For He^- the nuclear charge is $Z$=2. The total spin ${\hat{S}}$ and $z$-parity $\hat{\Pi}_{z}$ are also conserved quantities. The spectroscopic notation $\nu{}^{2S+1}M^{\Pi_{z}}$ is used to mark the states, where $\Pi_{z}$ denotes the $z$ parity eigenvalue $(\pm)$, and the quantum number $\nu$ labels the degree of excitation. For states with the same symmetry, for the lowest energy states at $\nu=1$ the notation is ${}^{2S+1}M^{\Pi_{z}}$. We always consider states with $\nu=1$ and $S_{z}=-S$ assuming they correspond to the lowest total energy states of a given symmetry in a magnetic field.

The assertion on the stability of the molecular ion ${\rm He}_{2}^{+}$ in strong magnetic fields requires a full understanding of the Helium atom and other Helium species in the presence of a strong magnetic field. It is known that the Helium atom exists for any magnetic field strength. Its ground state is realized by a singlet spin state $1{}^{1}S$ at zero and small magnetic fields $0\leq B\leq 0.75\,$a.u. For larger magnetic fields the ground state is realized by the fully polarized spin triplet state $1{}^{3}(-1)^{+}$ Becken and Schmelcher (2000). Despite the fact that the Helium atom in magnetic fields is one of the most studied systems, all such studies, we are familiar with, are limited to magnetic fields up to $B=100\,$a.u. (see, e.g. Becken and Schmelcher (2000)). For higher magnetic fields (up to the Schwinger limit $B_{\rm rel}=4.414\times 10^{13}\,$G $\sim 18783\,$a.u.) the relativistic corrections to the energy seem to be relatively small. Thus, we extend the study of the Helium atom up to the largest magnetic field $B=10^{4}\,$a.u. considered in the present study. In particular, a full understanding of the ${\rm He}$ atom in the spin triplet (ground) state ${1}^{3}(-1)^{+}$ at magnetic fields $10000\geq B\geq 100\,$a.u. is necessary.