Astrophys Space Sci (2007) 308: 267–277 DOI 10.1007/s10509-007-9337-7

O R I G I N A L A RT I C L E

Molecular systems in a strong magnetic field How atomic–molecular physics in a strong magnetic field might look like Alexander V. Turbiner

Received: 10 July 2006 / Accepted: 13 October 2006 / Published online: 21 March 2007 © Springer Science+Business Media B.V. 2007

Abstract Brief overview of one-two electron molecular systems made out of protons and/or α-particles in a strong magnetic field B ≤ 4.414 × 1013 G is presented. A particular emphasis is given to the one-electron exotic ions 3+ H++ 3 (pppe), He2 (ααe) and to two-electron ions H+ 3 (pppee),

He++ 2 (ααee).

Quantitative studies in a strong magnetic field are very complicated technically. Novel approach to the few-electron Coulomb systems in magnetic field, which provides accurate results, based on variational calculus with physically relevant trial functions is briefly described. Keywords Molecular ions (traditional and exotic) · Strong magnetic field PACS 31.15.Pf · 31.15-p · 31.10.+z · 32.60.+i · 97.10.Ld

Our goal is to present a brief review of the properties of 1e-atomic-molecular systems (both traditional and exotic (marked by bold)) in a magnetic field: 2+ 3+ H, H+ 2 , H3 , H4 ,

(HeH)2+ ,

(H−He−H)3+ ,

(He−H−He)4+ ,

He+ , He3+ 2 ,

A.V. Turbiner () Instituto de Ciencias Nucleares, UNAM, Mexico DF 04510, Mexico e-mail: [email protected]

(the list is complete for all magnetic fields in 0 ≤ B ≤ 4.414 × 1013 G) and 2e-systems 2+ H− , H2 , H+ 3 , H4 ,

(HeH)+ , He, He2+ 2 , (this list is incomplete so far) made from protons and/or αparticles. In general, a relevance of exploration of atomic-molecular physics in a strong magnetic field is motivated by a discovery of neutron stars which is characterized from one side by a strong surface magnetic field and from another side by the atmosphere possibly made from atomic-molecular compounds. Recent discovery by Chandra X-ray spatial observatory in the soft X-ray spectrum of the isolated neutron star 1E1027.4-5209 of two absorption features (Sanwal et al. 2002) pushed ahead drastically these studies. A present status of the 1e-molecular systems in a strong surface magnetic field is reviewed in Turbiner and Lopez-Vieyra (2006). What does it mean ‘strong magnetic field’? A magnetic field of order or higher than the characteristic atomic magnetic field for which the Larmor radius is equal to the Bohr radius B = B0 ≡

m2e e3 c = 2.3505 × 109 G. 3

Where a non-relativistic consideration can be used? For magnetic fields which are smaller than the Schwinger limit Brel —the magnetic field for which the electron cyclotron energy is equal to the electron mass B ≤ Brel =

m2e c3 = 4.414 × 1013 G. e

268

Hence, for magnetic fields 0 ≤ B  Brel the Schroedinger equation can be used. What is the Born–Oppenheimer (BO) approximation (of zero order)? Protons and/or α-particles are assumed infinitely massive. Corrections to BO due to finite nuclear mass(es) can be estimated by the lowest vibrational and rotational energies with a probable contribution coming from the cyclotron energies of the nuclei. Estimates show that these corrections appear to be of the same order of magnitude for all studied magnetic fields. They grow as a magnetic field increase but do not exceed 10–20% for binding energies for the highest B ≤ Brel . Thus, the numerical results obtained using the Schroedinger equation in BO approximation and presented below, in physics applications should be slightly modified by taking into account the relativistic corrections and the finite-mass effects. However, the existing accuracies in observational data do not require such a modification. Why the problem is so difficult? Mostly, it is due to the following reasons: • Highly-non-uniform asymptotics of the potential at large distances: in some spacial directions it grows ∝ r 2 , in others it vanishes. • It is a problem of several centers. • A posteriori we know that we deal with a problem of weakly-bound states, Ebinding /Etotal  1, 13 −2 (e.g. for H+ 2 at B = 10 G this ratio is  10 ).

In general, atomic units are used throughout the article if different is not stated. The energies and potentials are measured in Rydbergs.

Method – Variational Calculation. – Simple and unique trial function applicable for the whole range of accessible magnetic fields (0–4.414 × 1013 G which can lead to a sufficiently high accuracy in total energy.

How to choose trial functions? (see e.g. Turbiner 1984) – Physical relevance (as many as possible physics properties should be encoded even for a cost of simplicity). – Mathematical (computational) simplicity should not be a guiding principle. – Resulting perturbation theory should be convergent (see below).

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Variational calculus in a framework of perturbation theory (see Turbiner 1984) For chosen square-integrable Ψtrial let us find a trial potential Vtrial =

∇ 2 Ψtrial , Ψtrial

Etrial = 0.

(1)

Hence, we know the Hamiltonian for which the normalized Ψtrial is an eigenfunction Htrial Ψtrial = [p 2 + Vtrial ]Ψtrial = 0,

(2)

then  Evar =  =

∗ Ψtrial H Ψtrial ∗ ψtrial Htrial Ψtrial +   

 =0+



∗ Ψtrial (H − Htrial )Ψtrial

=0

∗ Ψtrial (V − Vtrial )Ψtrial = E0 + E1 .

(3)

Hence, – The variational energy is a sum of the first two terms of a certain perturbation series with perturbation potential (V − Vtrial ). – Choosing ψtrial appropriately we can obtain the convergence of the perturbation series (in order to get convertrial gence the ratio | V V−V | should be bounded if in addition trial this ratio tends to zero at large distances it leads to faster convergence) and if it is so the rate of convergence can be made as fast as possible (sometimes, it is reached by making a minimization of the energy functional with respect of parameters of ψtrial , however, it is not always the case that minimization leads to an increase the rate of convergence.). – How to calculate E2 in practice?—in general, it is unsolved problem yet, the only exception is one-dimensional case. Example Hydrogen atom in a magnetic field (ground state) In the excellent review by Garstang (1977) the physics of the problem is described, while the most accurate calculations at present are done in Kravchenko et al. (1996). If the magnetic field B is directed along z-axis the problem is characterized by the potential 2 B2 2 V =− + ρ , r 4

ρ2 = x2 + y2.

(4)

Let us choose a trial function ψ0 = exp(−αr − βBρ 2 /4),

(5)

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where α, β are physically meaningful variational parameters: α measures the charge (anti)-screening due to a magnetic field presence, β “measures” a magnetic permittivity. The trial potential is V0 =

Δψ0 2α β 2 B 2 2 αβB ρ 2 + ρ + , =− ψ0 r 4 2 r

(6)

and the reference point for energy is E0 = −α 2 + βB.

(7)

Variational energy gives relative accuracy ∼10−4 in total energy for 0 < B < 4.414 × 1013 G comparing to an accurate calculation!

Fig. 1 The geometrical setting for the (ppe) system

Remark (see Potekhin and Turbiner 2001) The trial function ψ0 = exp (−ϕ),

(8)

with ϕ=



α 2 r 2 + (γ1 r 3 + γ2 r 2 ρ + γ3 rρ 2 + γ4 ρ 3 ) + β 2 B 2 ρ 4 /16



where α, γ1...4 , β are parameters, gives relative accuracy ∼10−7 in total energy for magnetic fields 0 < B < 4.414 × 1013 G. The binding (ionization) energy Eb = B − ET grows with a magnetic field increase in quite drastic way. For instance, Eb (found variationally using ψ0 ) is equal to H:

where α1 , β1 , R are variational parameters. It is a product of the Heitler–London function and the lowest Landau orbital (modified by β1 ). This function is the exact eigenfunction of the lowest energy in the potential

Eb (10 000 a.u.) = 27.95 Ry,

He+ :

Eb (10 000 a.u.) = 78.43 Ry

which is many times larger than in the field-free case.

PART I 1 One-electron molecular systems (i) (ppe) system H+ 2 molecular ion (Parallel Configuration) The potential which describes the (ppe) system in a magnetic field with infinitely massive protons on the same magnetic line (see Fig. 1) is given by V =−

2 2 2 B 2ρ2 , − + + Bm + r1 r2 R 4

(9)

where ρ 2 = x 2 + y 2 and m is magnetic quantum number. Trial Functions for H2+ (ground state, m = 0) I. (r1 +r2 ) −β1 Bρ /4 ψ1 = e−α1  e  , 2

Heitler−London

Landau

(10)



B 2ρ2 + 2α1 2 − β1 B 4  1 1 + . + 2α1 2 n1 · n2 + α1 β1 Bρ 2 r1 r2   

V1trial = −2α1

1 1 + r1 r2

+ β1 2

(11)

V −Vtrial

At α1 = β1 = 1 the Coulomb singularities and Harmonic Oscillator behavior at large distances are reproduced exactly. At α1 , β1 = 1 (anti)screening of the nuclear charges and of the magnetic permittivity appear, respectively. We assume that the modified Heitler–London approximation can give a significant contribution for internuclear distances around the equilibrium. It will be verified a posteriori. Such a trial function describes a situation when the electron is attached to both charged centers assuming a type of coherent interaction. One can call it a ‘covalent’ coupling of the system. II. ψ2 = (e−α2 r1 + σ e−α2 r2 ) e−β2 Bρ   

2 /4

,

(12)

Hund−Mulliken

where α2 , β2 , R are variational parameters. Such a trial function describes a situation when the electron is attached to either one charged center or another assuming a type of incoherent interaction. One can call it a ‘ionic’ coupling of the system, H + p. We assume that this function can give a significant contribution for large internuclear distances. It will be verified a posteriori. In order to describe intermediate region between two domains R Req and R  Req we use two types of interpolations.

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Fig. 2 Electronic density |Ψ |2 for H+ 2 in parallel configuration

III-1. Non-linear Interpolation (simplest) ψ31 = (e−α3 r1 −α4 r2 + σ e−α3 r2 −α4 r1 ) e−β3 Bρ   

2 /4

,

(13)

Guillemin−Zener

where α3 , α4 , β3 , R are variational parameters. If α3 = α4 then ψ31 → ψ1 , if α4 = 0 then ψ31 → ψ2 . It realizes “ionic ↔ covalent coupling” interpolation which is verified a posteriori. III-2. Linear Interpolation ψ32 = A1 ψ1 + A2 ψ2 .

(14)

IV. Superposition of the two kinds of the interpolation ψ4 = A31 ψ31 + A32 ψ32 .

(15)

Conclusion: – This single function describes the system in the range of magnetic fields up to B = 4.414 × 1013 G. – Till very recently, this 10-parametric trial function ψ4 led to the lowest total energies for the ground state compared with numerous previous calculations for B > 1010 G up to B = 4.414 × 1013 G. Now the most accurate calculations are due to Vincke and Baye (2006). – For B  1010 G: Relative accuracy ∼ 10−5 in binding energy (by making a comparison with Vincke and Baye 2006). – A simple modification of ψ4 allows to study excited states 1σu , 1πg,u , 1δg,u etc.—the lowest states of different magnetic numbers and parities. Physical phenomenon: For B ∼ 5 × 1011 G the coupling changes from ‘ionic’ (incoherent) type to ‘covalent’ (coherent) type (see Fig. 2) (for further details see Turbiner and Lopez-Vieyra 2006). (ii) (ppe) system H+ 2 molecular ion (Inclined Configuration, see Fig. 3)

Fig. 3 The geometrical setting for the (ppe) system (inclined configuration)

Physics (see for references and details a review Turbiner and Lopez-Vieyra 2006): • Parallel configuration Θ = 0 is optimal for all studied magnetic fields (total energy takes a minimal value), see e.g. Fig. 4. 11 • H+ 2 does not exist for large inclinations at B > 10 G (total energy curve has no minimum at finite R), see Fig. 5. • H+ 2 is stable for all B and the most bound 1e-system made from protons for B  1013 G: H+ 2  H + p. Binding (ionization) energy of H+ 2 grows with a magnetic field increase. For example, Eb (10 000 a.u.) = 45.80 Ry, at Req = 0.118 a.u. is very large being almost twice larger than one for the H-atom (see above). Transition energy between the ground state and the lowest excited state also grows with a magnetic field increase reaching at 10 000 a.u. ΔE(1σg → 1πu ) = 11.73 Ry.

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Fig. 6 The geometrical setting for the (pppe) system (linear parallel configuration) Fig. 4 Total energy ET of the (ppe)-system as function R for different inclinations at B = 1 a.u.

of the positive parity 1πu , 1δg of the H2+ 3 ion can exist for B  1011 G. In general, the ion has a finite lifetime—it can decay + H2+ 3 → H2 + p,

however, a process H2+ 3  H + p + p, is always prohibited. For B  3 × 1013 G the first decay also becomes prohibited, H2+ 3 is stable system which is even the most bound 1e-system made from protons. Binding (ionization) energy of H2+ 3 is Eb (10 000 a.u.) = 45.41 Ry, eq

at R± = 0.130 a.u., it is very large in comparison with ionization energy for the H atom but it is comparable with H+ 2. Transition energy between the ground state and the lowest excited state at 10 000 a.u. is

Fig. 5 H+ 2 -ion: domains of existence ↔ non-existence

(iii) (pppe) system H2+ 3 : linear parallel configuration (The molecular axis and magnetic line coincide) The potential has a form V=

ΔE(1σg → 1πu ) = 12.78 Ry,

2 2 2 2 2 B 2ρ2 2 , + + − − − + R− R+ R− + R+ r1 r2 r3 4 (16)

(for a geometrical settings see Fig. 6). For B < 1011 G the total energy E(R+ , R− ) of the (pppe) system displays no minimum for finite R+/− , at most, a slight irregularity. However, at B  1011 G the total energy E(R+ , R− ) has explicitly pronounced minimum for finite R+ = R− which is stable towards small deviations from linearity. Hence, the parallel configuration is optimal. Thus, at B  1011 G the system (pppe) has a bound state, which manifests existence of the exotic molecular ion H2+ 3 (Turbiner et al. 1999). Furthermore, the excited states

being comparable with one for H+ 2. (iv) (pppe) system H2+ 3 : triangular configuration (The magnetic field is perpendicular to the equilateral triangle formed by protons) At B < 108 G the total energy of (pppe) in a spacial configuration has no minimum (or even does not display any irregularity) at finite interproton distances. However, at 108 G  B  1011 G the total energy E(R) has a wellpronounced minimum with protons in a triangular equilateral configuration at finite size R of a triangle side (LopezVieyra and Turbiner 2002) (see Fig. 7). This state of H2+ 3 is metastable (it has a finite lifetime) if the protons are externally supported in the plane perpendicular to the magnetic field direction. Otherwise, it is unstable. The minimum disappears at B > 1011 G, hence H32+ in triangular configuration does not exist anymore. This state can be considered

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Fig. 9 The geometrical setting for the (αpe) system Fig. 7 The geometrical setting for the (pppe) system (triangular configuration)

(HeH)(2+) . Also this systems can exist in two excited states 1π, 1δ. • For B  1013 G the ion (HeH)(2+) becomes stable: (HeH)(2+)  He+ + p. • Parallel Configuration is always optimal (see Fig. 9). Binding (ionization) energy is equal to

Fig. 8 (ppppe) in linear parallel configuration

as a precursor to linear configuration. No more stable or metastable spatial configurations of protons are found! (v) (ppppe) system H3+ 4 molecular ion (Parallel Configuration) Surprisingly, at B  1013 G the system (ppppe) has a minimum in total energy in linear parallel configuration at finite internuclear distances (see Fig. 8). It manifests the ex(3+) istence of the molecular ion H4 as metastable state. It can decay + H3+ 4 → H2 + p + p,

2+ H3+ 4 → H3 + p,

but H3+ 4  H + p + p + p. Binding (ionization) energy (see Turbiner and Lopez-Vieyra 2006) Eb (3 × 10 G) = 38.42 Ry, 13

is sufficiently large. Its excited state 1πu also exists but no more excited states are found. (vi) (αpe) system (HeH)2+ molecular ion (Parallel Configuration) • For B < 1012 G the system (αpe) is not bound. • For B  1012 G the system (αpe) has a bound state manifesting the possible existence of the molecular ion

Eb (10 000 a.u.) = 77.30 Ry, at Req = 0.142 a.u., it is almost twice larger then one for the 2+ H-atom and the hydrogenic ions H+ 2 , H3 being also larger than one for He+ . Transition energy between the ground state and the lowest excited state at 10 000 a.u. is ΔE(1σ → 1π) = 20.80 Ry, (for details see Turbiner and Lopez-Vieyra 2006, 2007). (vii) (ααe) system He3+ 2 molecular ion (Parallel Configuration) • For B < 2 × 1011 G the system (ααe) does not exist • For B  2 × 1011 G the system (ααe) is bound manifest(3+) ing the existence of the molecular ion He2 , and even two excited states of positive parity 1πu , 1δg do also exist (3+) • For B  1012 G ion He2 becomes stable: (3+)

He2

 He+ + α,

• Parallel Configuration is always optimal. Binding (ionization) energy Eb (10 000 a.u.) = 86.23 Ry, +

(cf. [EbHe (10 000 a.u.) = 78.43 Ry]) at Req = 0.150 a.u. It is almost twice larger in comparison with one for the atom 2+ + H and hydrogenic ions H+ 2 , H3 , and also larger than He .

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Fig. 10 The geometrical setting for the (ααe) system

Transition energy between the ground state and the lowest excited state at 10 000 a.u. is equal to ΔE(1σg → 1πu ) = 24.69 Ry, (for details see Turbiner and Lopez-Vieyra 2006, 2007), it is the largest among transition energies for one-electron systems. It is discovered the striking approximate relation between the binding energies of the most bound one-electron systems made from α-particles (i) and made from protons (ii): He+ ,He2

(3+)

Eb

H+ ,H2+ 3

≈ 2Eb 2

,

(17)

which holds for 1011 G < B < 1014 G. • For 1011 G < B < 1012 G in l.h.s. Eb is the binding energy of the He+ atomic ion, otherwise Eb is the binding energy of the exotic ion He3+ 2 . • For 1011 G < B < 1013 G in r.h.s. Eb is the binding energy of H+ 2 , otherwise Eb is the binding energy of the exotic ion H2+ 3 .

13 – H2+ 3 has the lowest Etotal for B  10 G among 1e systems made from protons. – The Hydrogen atom has the highest total energy being the least bound 1e system made from protons for 0 < B  4.414 × 1013 G. (4+) for B > 4.4 × – Possible existence of the system H5 1013 G (at ∼1013 G); but a reliable statement requires a consideration of relativistic corrections and finite-mass effects. It would be the longest 1e hydrogenic chain. has the lowest to– For B  1012 G the exotic He3+ 2 tal energy among systems made from protons and/or αparticles. 2+ – At B ∼ 3 × 1013 G for the ions H+ 2 and H3 in optimal configuration the binding energies ≡ ionization energies coincide, both are equal to ∼700 eV, while for the He3+ 2 -ion in parallel configuration the ionization energy is ∼1400 eV. It corresponds to the energies of absorption features observed in the radiation from 1E1027.4-5209 Sanwal et al. (2002). – Many more exotic 1e systems may occur at the Schwinger limit B ∼ 4.414 × 1013 G and beyond.

For molecular systems we are not aware about any quantitative reliable studies of radiation transitions, both boundbound and bound-free ones. • Technical observation: Surprisingly, many quite sophisticated methods allow to find 1, 2, 3 significant digits in binding energy only. For example, the Hartree–Fock method used by E. Salpeter and collabo11 G gives a sinrators (see Lai et al. 1992) for H+ 2 at 10 gle significant digit only. Usually, serious difficulties occur when attempting to go beyond those 1–3 significant digits, to higher accuracy. It may need narrow-specialized methods to employ and much efforts. It was implicitly demonstrated in the numerous papers by P. Schmelcher and co-authors and, finally, in Vincke and Baye (2006).

3 Physics of 1e-systems in a magnetic field 2 Summary One-electron linear systems (for details see Turbiner and Lopez-Vieyra 2006) Optimal configuration for all linear (3+) (3+) 2+ , (HeH)2+ and He2 is parmolecular ions H+ 2 , H3 , H4 allel: heavy centers are situated along magnetic field line (when exist). When magnetic field grows: 2+ 3+ 2+ and He3+ – Binding energy of H, H+ 2 , H3 , H4 , (HeH) 2 grows (whenever the given system exists). 2+ 3+ 2+ and – Natural size of the systems H+ 2 , H3 , H4 , (HeH) He3+ 2 decreases. 13 G among 1e – H+ 2 has the lowest Etotal for 0 < B  10 systems made from protons.

Any quantal charged particle moves in a magnetic field almost freely inside of a channel situated along a magnetic line of the average transverse size which is defined by the Larmor radius, ∼B−1/2 . The walls of this channel are “soft”, the potential grows quadratically in transverse size ρ. Placing several charged particles inside of this channel one can check the electrostatic stability of the system calculating the Coulomb energy (see Fig. 11). One of the simplest systems of such a type is one when there are two heavy charges Z and the electron situated in the middle of them. It is easy to find that the Coulomb energy of this system is negative for charges Z = 1, 2, 3 and for any distance between heavy charges which implies a stability. A natural question is for what size of the channel the transverse fluctuations do not

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A method of construction of trial functions described above we applied to a study of the ground state of H2 , H+ 3 molecules in Born-Oppenheimer (BO) approximation. We begin from overview of the field-free case.

5 H2 (ground state) EBO = −2.3469 Ry (James and Coolidge 1933, 15 parameters). EBO = −2.3478 Ry (Heidelberg group ’01, >200 noncentered Gaussian orbitals). EBO = −2.3484 Ry (see Turbiner and Guevara 2007, 14 parameters). EBO = −2.3489 Ry (record calculations, 1000 James and Coolidge type functions). 6 H+ 3 (lowest linear spin-triplet state)

Fig. 11 1e molecular systems: a qualitative picture

ruin such a picture. As for traditional ion H+ 2 this picture holds for any transverse size—the system (ppe) is bound for any magnetic field strength. A first critical transverse size is ∼0.1 a.u.—for this transverse size three exotic twoand three-center systems begin to occur: (HeH)2+ , He3+ 2 , H2+ and then they continue to exist for smaller transverse 3 sizes. They appear at first as metastable systems but for the transverse size 0.01 a.u. become stable. Surprisingly, this transverse size is also the next critical transverse size. At the transverse size ∼0.01 a.u. five more exotic two-, three- and four-center systems begin to occur: (Li H)3+ , Li5+ 2 , (H–He– 3+ 3+ 4+ H) , (He–H–He) , H4 , then they continue to exist for smaller transverse sizes.

PART II 4 Two-electron molecular systems Two-electron systems in a magnetic field is a subject which was explored in a very little extend. It is related with a fact that the studies of these systems are quite complicated technically. To our opinion even the simplest atomic systems H− , He are not fully understood (see e.g. Al-Hujaj and Schmelcher 2003 and references therein) although it is undoubted that both systems exist for all accessible magnetic fields. One can say that the simplest molecule H2 only was a subject of quite intense studies. However, a situation is far to be clear.

EBO = −2.2284 Ry (Schaad et al.’74, Configuration Interaction (CI) method (see Turbiner et al. 2006a for reference)). EBO = −2.2297 Ry (see Turbiner et al. 2006a, 23 parameters). EBO = −2.2322 Ry (Preiskorn et al. 91, CI + James and Coolidge type functions (see Turbiner et al. 2006a for reference)). In Turbiner et al. (2006a, 2006b) electronic correlation appears in explicit form exp(γ r12 ) in trial functions, where γ is a parameter. A difficult technical problem of the multidimensional integration with high precision which in the past always was tried to avoid by everybody was successfully resolved.

7 Physics of 2e-systems in a magnetic field It is well-known that in absence of magnetic field the state of the lowest energy is the spin-singlet, the electron spins are antiparallel. It seems absolutely natural that for a sufficiently strong magnetic field the state of the lowest energy (if bound) is the spin-triplet, with both electron spins are antiparallel to the magnetic field direction. Hence, for a certain magnetic field there is a transition from spin-singlet to spintriplet state. Therefore quantum numbers of the ground state are changed. However, a situation might be more complicated than described. If we neglect inter-electron interaction then for spin-singlet case the electrons can have the same quantum numbers except for the spin projection. A transition to spin-triplet state can occur without a change of the electron quantum numbers except for the spin projection if electrons are situated sufficiently far away from each other and the Pauli repulsion can be non-essential. Of course, if

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the electrons are close to each other a transition singlet— triplet is accompanied by a change of quantum numbers of electron(s). It is exactly what happens in the case of the Helium atom—for zero and small magnetic fields the ground state is 1 σ and then for a certain magnetic field the state 3 π becomes the ground state Al-Hujaj and Schmelcher (2003). It might be different for the case of molecules—it might exist a domain in magnetic fields where the lowest energy state is 3 Σ. It is known that at very large magnetic fields the transverse size of the electronic cloud is very small, of the order of the Larmor radius. The longitudinal size also shrinks as a magnetic field grows, ∼1/log B. It leads naturally to a conclusion at a sufficiently large magnetic field for any two-electron Coulomb system the lowest energy state should be 3 Π —the spin-triplet state with the total magnetic quantum number equals to −1 Kadomtsev and Kudryavtsev (1971); Ruderman (1971). What remains to be answered is two quantitative questions: from what magnetic field strength the state 3 Π is the ground state and does a domain in B exist where the state 3 Σ is the state of the lowest energy. As for the latter question it is found that for two two-electron atomic-like systems: H− and He this domain is absent (see Al-Hujaj and Schmelcher 2003 and references therein). We follow a physics picture of the channel which surrounds the magnetic line. Let us “place” inside of the channel several heavy, positively charged particles and two electrons. One can check the electrostatic stability of the system by calculating the Coulomb energy. The simplest systems of such a type are made of several protons and two electrons. It is easy to check that the Coulomb energy is negative for the case of 1, 2, 3, 4, 5 protons and it remains true for any distances between protons (see Fig. 12). It assumes the electrostatic stability of these systems. Natural questions are (i) for what size of the channel the transverse fluctuations do not ruin such a qualitative picture and (ii) how to estimate equilibrium distances. We do not know how to answer these questions qualitatively although quantitatively it can be answered by solving the corresponding Schroedinger equation. (i) (pppee) system + H3 (Turbiner et al. 2006a, first detailed study) The Hamiltonian is H=

2 

=1

+

−∇ 2 +



B2 2 ρ − 4

=1,2 κ=a,b,c

2 + B(Lˆ z + 2Sˆz ), R+ + R−

2 2 2 2 + + + r κ r12 R+ R− (18)

Fig. 12 2e-molecular systems: a qualitative picture

Fig. 13 The H+ 3 molecular ion in parallel configuration in a uniform constant magnetic field B = (0, 0, B)

(for a geometrical setting see Fig. 13), where Lˆ z = lˆz1 + lˆz2 and Sˆz = sˆz1 + sˆz2 are the z-components of the total  angular momentum and total spin, respectively, and ρ =

x 2 + y 2 .

Trial function: ψ (trial) = (1 + σe P12 ) × (1 + σN Pac )(1 + σNa Pab + σNa Pbc ) |m|

× ρ1 ei mφ1 eγ r12 −β1 Bρ1 /4−β2 Bρ2 /4 2

2

× e−α1 r1a −α2 r1b −α3 r1c −α4 r2a −α5 r2b −α6 r2c ,

(19)

where σe = ±1 stands for spin singlet (S = 0) and triplet states (S = 1), respectively. For S3 -permutationally symmetric case σN = σNa = ±1. Pac interchanges the two extreme protons a and c, and α1−6 , β1−2 and γ are variational parameters. The operators P12 interchanges electrons (1 ↔ 2). One can consider possible degenerations of ψ (trial) which appear when some α’s are made equal and/or vanish. Then to take a linear superposition of ψ (trial) and its degenerations as a new trial function.

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Fig. 14 Ground state evolution for the H+ 3 -ion in parallel configuration as a function of the magnetic field strength

Fig. 15 Ground state evolution for the He2+ 2 -ion in parallel configuration as a function of the magnetic field strength where 1 Σg is metastable, 3 Σu is unbound, 3 Πu is strongly bound

The results for H+ 3: • For any magnetic field the H+ 3 ion is stable. • For magnetic fields B ≥ 5 × 108 G the optimal configuration is the linear, parallel, symmetric (R+ = R− ) and the system is stable towards all small deviations. For large magnetic fields B ≥ 5 × 1010 G the ground state is 3 u while for the intermediate fields the ground state is given by the weakly bound 3 Σu state (see Fig. 14). • If for magnetic fields B ≤ 5 × 108 G a linear parallel configuration is kept externally a minimum is developed at R+ = R− . However, the system remains unstable towards small deviations from linearity. The true ground state in this domain does exist but it corresponds to an equilateral triangular configuration. At B = 10 000 a.u. the total energy of the ground state of H+ 3 is equal to 3 ET (H+ 3 ( Πu )) = −95.21 Ry, eq

at R± = 0.093 a.u. It coincides with the minus double ionization energy and is significantly smaller than the total energy of the ground state of the H2 molecule (see below) ET (H2 (3 Πu )) = −71.39 Ry, as well as the sum of the total energies of H+ 2 (1πu ) and H(1s) ET (H+ 2 (1πu ) + H(1s)) = −62.02 Ry. It is worth noting that the dissociation energy for H+ 3 → H2 + p is quite large, 23.82 Ry. The transition energy from the ground state to the lowest excited state 3 Δg is equal to ΔE(3 Πu → 3 Δg ) = 7.76 Ry. (ii) (ααee) system He2+ 2 (see Turbiner and Guevara 2006, the first study) (parallel configuration, the lowest excited states) The results for He2+ 2 (see Fig. 15):

• For all magnetic fields where the He2+ 2 ion exists the parallel configuration is optimal, 2+ + + • He2+ 2 is metastable at B ≤ 0.85 a.u.: He2 → He + He • He2+ 2 is stable and strongly bound at B ≥ 1100 a.u. with 3 Π as the ground state u • He2+ 2 does not exist at surprisingly large domain 0.85 ≤ B ≤ 1100 a.u. At B = 10 000 a.u. the total energy of the ground state 2+ u of He2 is

3Π

ET (3 Πu ) = −174.51 Ry, with Req = 0.106 a.u. while the total energy of two Helium ions with the parallel electron spins is ET (He+ + He+ )  −156.85 Ry for the state (1s1s), = −137.26 Ry for the state (1s2p−1 ), where in brackets the quantum state of the first and second Helium ion, respectively, are indicated. It is worth noting that ET (3 Πu ) is almost twice less than the total energy of He3+ 2 (1σg ) + e: ET (He3+ 2 (1σg ) + e) = −86.23 Ry. The transition energy from the ground state 3 Πu to the lowest excited state 3 Δg is equal to ΔE(3 Πu → 3 Δg ) = 13.87 Ry, it is small comparing to ionization-dissociation energies. (iii) (ppee) system H2 (ground state (Turbiner 1983 (first calculation), . . . Heidelberg group ’90–’03, in particular, Detmer et al. 1998, for a review and references see e.g. Al-Hujaj and Schmelcher 2003).

Astrophys Space Sci (2007) 308: 267–277

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(He–H–He)3+ , He4+ 3 , etc. We do not think it is a moment to draw a conclusion about physics of 2e molecular systems in a strong magnetic field. So far any concrete applications are obstructed by absence of analysis of radiation transitions. Acknowledgements The author brings his sincere gratitude to J.C. López Vieyra and N.L. Guevara with whom he shared a pleasure to study this beautiful physics, and to G.G. Pavlov for the encouragement to write a present mini-review. This work was supported in part by FENOMEC and PAPIIT grant IN121106. Fig. 16 Ground state evolution for the H2 molecule in parallel configuration as a function of the magnetic field strength where 1 Σg is stable, 3 Σ is unbound, 3 Π is strongly bound u u

The results: • When H2 exists the parallel configuration is optimal. • The ground state evolves from 1 Σg at B  0.18 a.u. to unbound 3 Σu (Req tends to infinity, the system appears as two separated H-atoms) and then at B  15.6 a.u. to strongly bound 3 Πu state (see Fig. 16 and Al-Hujaj and Schmelcher 2003, the numbers from Turbiner and Guevara 2007). • When H2 exists the system is always stable, however, for all magnetic fields the total energy of the ground state is ET (H2 ) > ET (H+ 3 ). Due to a presumed importance of this system in a strong magnetic field several research teams made studies, many striking qualitative effects are predicted. However, it must be emphasized that due to the technical complexity of the problem the accuracy of obtained results is hard to be estimated. For example, our analysis says that the total energy of the 3 Πu state at B = 1011 G obtained in the Hartree–Fock method Lai et al. (1992) gives correctly not more that two significant digits. As for the 1 Σg state at B ≤ 0.2 a.u. not more than three significant digits in the binding energy are found correctly in the Gaussian orbital method employed by the Heidelberg group although it was claimed the six digit accuracy. 3+ Many more 2e systems should be studied: H2+ 4 , H5 . . . + (2e hydrogenic linear chains?), (HeH) , (H–He–H)++ ,

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