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H_3^(2+) molecular ion in a strong magnetic field: a triangular configuration

The bound state in the system of three protons and an electron (pppe) under a homogeneous strong magnetic field where the protons are situated in the vertices of an equilateral triangle perpendicular to the magnetic field lines is found. It is shown that for magnetic fields B = 10^11 - 4.414x10^13 G the potential energy curves as a function of the internuclear distance R have an explicit minimum. For all magnetic fields studied, the binding energy of the triangular configuration is less than the binding energy of the linear parallel configuration (A. Turbiner et al. JETP Lett. 69, p. 844). In the contrary to the linear case, the binding energy decreases with a magnetic field growth, while the equilibrium internuclear distance slowly increases.

The bound state in the system of three protons and an electron (pppe) under a homogeneous strong magnetic field where the protons are situated in the vertices of an equilateral triangle perpendicular to the magnetic field lines is found. It is shown that for magnetic fields B = 10 11 − 4.414 × 10 13 G the potential energy curves as a function of the internuclear distance R have an explicit minimum. For all magnetic fields studied, the binding energy of the triangular configuration is less than the binding energy of the linear parallel configuration (A. Turbiner et al. JETP Lett. 69, p. 844). In the contrary to the linear case, the binding energy decreases with a magnetic field growth, while the equilibrium internuclear distance slowly increases.
Recently a quantitative study of the system of three protons and an electron (pppe) in a strong magnetic field gave theoretical evidence of the existence of a bound state for magnetic fields B > 10 11 G [1]. It manifests the existence of the exotic molecular ion H (2+) 3 . Many years ago it was indicated by Kadomtsev and Kudryavtsev [3], and Ruderman [4] the possible existence of linear exotic molecular systems in presence of a strong magnetic field. Due to the enormous Lorentz force acting on the electronic cloud, it shrinks in the direction transversal to the magnetic field lines leading to an effective quasi-one-dimensionality of the studied systems. This makes the electron-nuclei attraction more effective to compensate the Coulombic repulsion of nuclei. Such a fact suggests the possibility of the existence of exotic systems which do not exist without a magnetic field. It has been shown some time ago that a strong magnetic field can lead to the formation of linear hydrogenic chains H n , n > 2 situated along magnetic lines [4][5][6][7]. More recently the first quantitative study about the possible existence of an exotic bound state in the system of three protons and an electron (pppe) in a strong magnetic field was carried out [1] and it confirmed existence of H (2+) 3 ion. Although the linear parallel configuration (where protons are situated on a line parallel to the magnetic field line) for the system (pppe) seems to be optimal for very strong magnetic fields, other possible configurations have to be considered in order to have a better understanding of the properties of such systems.
In this article we explore the possibility of the existence of the molecular ion H , where the protons are situated in the vertices of an equilateral triangle perpendicular to a homogeneous magnetic field. We limit our present investigation to magnetic fields B = 10 11 − 4.414 × 10 13 G for which a non-relativistic approach is still valid (for a detailed discussion see [7] and references therein). Our study is restricted to the ground state within the Born-Oppenheimer approximation.
The present calculations are carried out in the framework of a variational method. Trial functions are chosen in accordance with the following criterion (see [9]- [14]): (i) the trial function Ψ t (x) should include all symmetry properties of the problem in hand; (ii) the trial function for the ground state should not vanish inside the domain where the problem is defined; (iii) the potential V t (x) = ∇ 2 Ψt Ψt , for which the trial function is an exact eigenfunction, should reproduce the original potential behavior near singularities as well as its asymptotic behavior. This prescription has been successfully applied to the study of the H + 2 ion in strong magnetic fields giving the more accurate results for B > 0 [8], as well as to the study of the linear parallel configuration of the H ions in a strong magnetic fields [1,2]. We consider a system of one electron and three identical infinitely-heavy centers of unit charge situated on the vertices of an equilateral triangle with internuclear distance R (which is the length of the triangle side) lying in a plane perpendicular to the magnetic field of strength B directed along the z axis, B = (0, 0, B).
The potential corresponding to the system we study is given by where the quantity ρ = x 2 + y 2 is the distance from the electron position to the z-axis, r 1,2,3 are the distances from the electron to the first, second and third centers, respectively, R is the distance between centers (see Fig. 1 for notations) andl z is the z-component of the angular momentum operator. The linear Zeeman effect term in the potential (Bℓ z ) can be dropped off since we use real trial functions 1 (see below). Spin Zeeman effects are neglected. Through the paper the Rydberg is used as the energy unit. For the other quantities standard atomic units are used. The trial functions we are going to use for the present problem are similar to those were exploited in [1] for the study of the linear parallel configuration. These functions were built according to the criterion described in [9]- [14]. They contain the basic features of Coulomb systems in a magnetic field, as well as the symmetry property under permutation of the three charged centers. In particular, it implies that the potentials corresponding to those functions reproduce the Coulomb-like behavior near the centers and the two-dimensional oscillator behavior in the (x, y) plane at large distances (see below).
The simplest of those functions is a function of the Heitler-London type multiplied by the lowest Landau orbital: (cf. Eq. (2) in Ref. [1]), where α 1 and β 1 are variational parameters. Since we consider the distance between centers as an extra variational parameter, it has in total three variational parameters. This function gives an adequate description of the covalent coupling of the system near equilibrium. As illustration we show the potential corresponding to this function: wheren i (i = 1, 2, 3) is the unit vector in the direction of the vector pointing from the position of the i-th center to the position of the electron. A second trial function is a Hund-Mulliken type function multiplied by the lowest Landau orbital: (cf. Eq. (4) in Ref. [1]). Here α 2 , β 2 are variational parameters. This function describes a ionic coupling of a hydrogen atom with two charged centers. Another suitable trial function which describes a ionic coupling between an H + 2 ion and a proton is given by (cf. Eq. (5) in Ref. [1]) where α 3 , β 3 are variational parameters. Since the H + 2 ion has the lowest total energy among the one-electron systems for B < ∼ 10 13 G [2], an important contribution coming from this trial function is expected. In order to include in a single trial function the different physical behavior near equilibrium and at large distances, appropriate interpolations of the trial functions (2), (4) and (5) are done: (i) A natural interpolation is given by a non-linear superposition of the form: (cf. Eq. (6) in Ref. [1]) where α 4 , α 5 , α 6 and β 4 are variational parameters, and the sum is over all permutations of the parameters {α 4 , α 5 , α 6 }.
If all parameters coincide (α 4 = α 5 = α 6 = α 1 ), the function (6) reduces to the Heitler-London type function (2). Function (6) also reduces to the Hund-Mulliken type wave function (4) when only one parameter is non-zero say α 4 = α 2 and α 5 = α 6 = 0. If two parameters are non-zero, and equal, say, α 4 = α 5 = α 3 , and α 6 = 0 it reduces to the trial function (5). When all parameters are different among them and different from zero the function (6) provides us with a 3-center modification of a Guillemin-Zener type function. The function (6) has in total five variational parameters (including the internuclear distance R as one of the parameters).
(ii) Another more immediate interpolation is given by a linear superposition of the functions (2), (4) and (5).
where A 1 , A 2 , A 3 are taken as extra variational parameters. Since Ψ 1,2,3 are not orthogonal, the parameters A 1,2,3 do not have the usual meaning of weight factors.
For the present calculations we use a linear superposition of the above interpolations (6), (7) given by: where A 1 , A 2 , A 3 , A 4 are considered again as extra variational parameters with no meaning of weight factors. The function (8) combines in a single function form, a suitable description of the system for the different physical regimes (near equilibrium and large internuclear distances for strong and very strong magnetic fields). Therefore we guess it should provide a relevant approximation for the ground state of the system. This was indeed the case of the linear parallel configuration of H (see Ref. [1]). This function has in total fourteen variational parameters. The variational procedure is carried out using the standard minimization package MINUIT from CERN-LIB. We use the integration routine D01FCF from NAG-LIB. All integrals are calculated with relative accuracy > ∼ 10 −8 . The results of our calculations are presented in Table I. For all considered magnetic fields B = 10 11 − 4.414 × 10 13 G we found a minimum in the potential energy curve as a function of the internuclear distance R (see below Fig. 3) indicating the formation of a bound state. For such magnetic fields the total energy of the triangular configuration is always larger than the total energy of the corresponding linear parallel configuration. Indeed this shows that for very strong magnetic fields the linear configuration is the most favored. However, a new striking unexpected phenomenon appears when one considers a triangular configuration. While for the linear configuration the binding energy 2 increases and the internuclear distance decreases as the magnetic field grows, the opposite is true for the triangular configuration. Namely, the binding energy is slowly decreasing and the internuclear distance is slowly increasing (!) both being almost constant. A similar behavior occurs with the longitudinal localization length of the electron |z| which also slowly increases in the triangular configuration with a magnetic field growth, opposite to what happens in the linear configuration where it decreases rather sharply. The transversal size of the electronic cloud ρ is very close to the corresponding cyclotron radius B −1/2 (a.u.) for both cases. We do not have a clear physical picture of this phenomenon and it needs further consideration.
A surprising feature is revealed if we study different trial functions. In Table II it is shown the variational results for the total energy E T for two quite arbitrarily chosen values of the internuclear distance, R = 0.45 and R = 2.527. For magnetic fields 10 11 < ∼ B < ∼ 10 12 G an adequate description of the system for internuclear distances R ≃ 0.45 is given by the trial function (5) modelling a ionic coupling, H + 2 +proton. In this case, the single function (5) gives the lowest total energy among the trial functions Ψ 1,2,3,4 . On the other hand, the trial function (2) modelling a covalent coupling gives the lowest total energy for R ≃ 2.527 giving a hint that a covalent coupling dominates when the internuclear distance is large compared to the equilibrium distance. Amazingly, for B > ∼ 10 13 G the variational energy seems to be insensitive to the specific form of the trial function (!).
The electronic density distribution |Ψ| 2 in the plane z = 0 is shown in Fig. 2. For all magnetic fields B = 10 11 − 4.414 × 10 13 G the electronic density distribution |Ψ| 2 exhibits a single maximum located at the center of the triangle (x = 0, y = 0). As the magnetic field grows the electronic cloud becomes more concentrated around x = 0, y = 0, while its distance to the protons is slowly increased.
From our results we can draw the conclusion that the exotic molecular ion H (2+) 3 can exist in the presence of a strong magnetic field in a triangular configuration of the charged centers which is unstable towards a decay mode to the linear configuration. In particular, in the present article we have shown that the molecular ion H (2+) 3 can also exist in the equilateral (fixed nuclei) triangular configuration perpendicular to the magnetic field.
The results of our variational study for the ground state show that, contrary to the case of the linear parallel configuration, the binding energy of the system decreases, and the internuclear distance increases, as the magnetic field grows from B = 10 11 G up to 4.414 × 10 13 G. A more extended study of this as well as other configurations is needed in order to have a better understanding of the properties of the exotic system H (2+) 3 . In particular, a proper contribution coming from the linear Zeeman effect term in (1) has to be taken into account. However, the fact that the linear Zeeman effect contribution vanishes in our consideration (which occurs when real trial functions are considered) looks relevant in the region of very strong magnetic fields B = 10 11 − 4.414 × 10 13 G where this term can be neglected anyhow.
Although in general it is not clear so far (see discussion in [7]) how an adiabatic separation of the electronic and the nuclear motion can be performed in the presence of a strong external magnetic field, our fixed-nuclei approach is a good starting point to study of the exotic molecular ion H (2+) 3 .

ACKNOWLEDGMENTS
The author wish to express his deep gratitude to Alexander Turbiner for introducing to the subject and for numerous valuable discussions.
This work is supported in part by DGAPA Grant # IN105296 (México). ion in a strong magnetic field. The total energy ET and the binding energy E b = B − ET , are in Ry. The equilibrium internuclear distance Req and the expectation values of the longitudinal and transversal localization length of the electron |z| and |ρ| are in a.u.. The corresponding quantities for the H (2+) 3 ion in the linear parallel configuration were taken from [1]. The energy of free electron and its cyclotron radius B −1/2 are also shown.