Few-Body Systems 38, 133–137 (2006) DOI 10.1007/s00601-005-0129-8

Non-Relativistic H12 -Molecule in a Strong Magnetic Field
R. Benguria1 , R. Brummelhuis2 , P. Duclos3 , S. P
erez-Oyarz
un4 , and P. Vytras5 1

2

3

4

5

Departamento de Fı´sica, Pontificia Universidad Cat
olica de Chile, Casilla 306, Santiago 22, Chile Birkbeck College, University of London, School of Economics, Mathematics and Statistics, 7-15 Gresse Street, London, UK Centre de Physique Th
eorique UMR 6207, Unit
e Mixte de Recherche du CNRS et des Universit
es Aix-Marseille I, Aix-Marseille II et de l’Universit
e du Sud Toulon-Var, Laboratoire affili
e a la FRUMAM, Luminy Case 907, F-13288 Marseille Cedex 9, France Instituto de Ciencias Basicas, Facultad de Ingenieria, Universidad Diego Portales, Av. Ejercito No. 441, Casilla 298-V, Santiago, Chile Katedra Matematiky, FJFI, CVUT, Trojanova 13, Praha 12000, Czech Republic

Received November 11, 2005; accepted November 30, 2005 Published online May 31, 2006; # Springer-Verlag 2006

Abstract. We show that under the influence of a strong uniform magnetic field the energy of the Hþ 2 -ion at the 0-th order Born-Oppenheimer approximation goes over into that of the corresponding united atom limit, Heþ : 1 Introduction Atoms and molecules in a strong uniform magnetic field of strength B will effectively behave like systems in one dimension, since the field will ‘‘freeze’’ the motion of the electrons perpendicular to the field into Landau orbitals. The electrons will only be free to move along the field-direction, under the influence of onedimensional effective potentials induced by the original Coulomb interactions. In the high field limit, these effective potentials are well-approximated by zero-range
-interactions, with a B-dependent coupling constant. This physical picture can be given a rigorous mathematical foundation for atoms and molecules having infinitely heavy nuclei aligned along the field direction, with the successive approximations holding true in the fairly strong sense of norm-convergence of resolvents, and explicit error bounds [1–5]. This can be used to draw rigorous conclusions, for
Article based on the presentation by R. Brummelhuis at the Fourth Workshop on the Dynamics and Structure of Critically Stable Quantum Few-Body Systems, MPIPKS, Dresden, Germany, 2005

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the original atomic or molecular Hamiltonian, from the
-model, which in oneelectron cases is elementarily solvable. We illustrate this by a detailed study of the Hþ 2 -molecule in a strong magnetic field, for which we prove that the equilibrium distance between the nuclei tends to 0 as the field strength tends to infinity, and the ground state energy tends to that of its united atom limit. 2 The Asymptotic Model We consider a non-relativistic one-electron homonuclear diatomic molecule with fixed nuclei of charge Z in a strong homogeneous magnetic field B ¼ B^z, where ^z is the unit vector in the z-direction. If the inter-nuclear distance is R, then the Pauli-Hamiltonian for the molecule, in atomic units, is
2
Z2 H ¼ 12
p  12r ^ B
þ   B  V þ ; R where V is the electron-nuclei potential Z Z
þ

; VðrÞ ¼





r  R ^z

r þ R ^z



2 2


ð1Þ

ð2Þ

and  ¼ ðx ; y ; z Þ the electron spin vector, given by the Pauli matrices. The conversion to the field strength in Gauss is done by multiplication of B by h3 ¼ ’ 2:35  109 G. In ref. [5], it was shown that atomic Hamiltonians B0 :¼ m2e e3 c= in strong magnetic fields can be approximated, in norm-resolvent sense, by a hierarchy of effective Hamiltonians describing one-dimensional atoms on the line. The machinery of ref. [5] is still applicable to the molecular case, provided the nuclear axes are taken parallel to B, to ensure that the total electron-angular momentum in the field direction is preserved (this is no longer true for arbitrary orientations). The simplest of the effective Hamiltonians of refs. [5, 4], giving the lowest-order approximation, is the
-Hamiltonian, which in the present case is given by   1 2 X R Z2 þ : ZL
z  ð3Þ h
¼ pz  2 2 R  pffiffiffi Here L ¼ LðBÞ :¼ 2Wð B=2Þ, W: ½e1 ; 1Þ ! R being the principal branch of the Lambert function, defined as the unique real solution of WðxÞeWðxÞ ¼ 1 which is positive for positive x; see, e.g., ref. [6]. Note that h
still depends on B, through L. We have that LðBÞ ’ log B as B ! 1. Under the reasonable assumption that the electron is in an s-state (this is not essential), h
will approximate H in the following sense: Let P0
be the orthogonal
2 projection onto the lowest Landau band of the ‘‘free’’ operator 12
p  12 r ^ B
with z-angular momentum m ¼ 0, and let P? 0 be the projection onto the orthogonal complement; P0 commutes with H, and h
has a natural interpretation as an operator on RanðP0 Þ. Let H
:¼ h
P0 þ HðBÞP? 0 . Then: Theorem 1 (compare ref. [5], Theorem 1.5). If d
ðÞ is the distance of  2 R to the spectrum of h
, then there exist positive constants c
, C
and B
, only depending

Non-Relativistic Hþ 2 -Molecule in a Strong Magnetic Field

135

on Z, such that if B  B
and c
L  d
ðÞ  L2 =4, then  is in the resolvent set of H, and L jjðH  Þ1  ðH
 Þ1 jj  C
: ð4Þ d
ðÞ2 The spectrum of HP? 0 turns out to be positive, and d
ðÞ > 0 will imply that  < 0, since the essential spectrum of h
already contains ½0; 1Þ. Theorem 1 allows us to deduce information about the negative bound states of H from those of h
. Eq. (4) may look strange as an approximation result, since L in the left-hand side goes to 1. However, the ground state energy of h
is of the order of cL2 in absolute value, and the same can then be shown to be the case for H, see below. In refs. [1–3], a re-scaled version of Theorem 1 was used. The
-model is explicitly solvable, and h
can be shown to have two eigenvalues,   1 Wðxex Þ 2 Z 2 2 e0 ¼ e0 ðR; L; ZÞ ¼  ðLZÞ 1 þ þ ; 2 x R   1 Wðxex Þ 2 Z 2 ð5Þ þ ; e1 ¼ e1 ðR; L; ZÞ ¼  ðLZÞ2 1 þ 2 x R where x :¼ RLZ; note that xex   e1 , for all x  0. The corresponding eigenfunctions can also be computed explicitly, cf. ref. [1]. The ground state energy of h
is e0 , and the molecule will bind iff inf R ½e0 ðR; L; ZÞ  eat  < 0, where eat ¼ Z 2 L2 =2, the ground state energy when the two nuclei are at infinite distance. The equilibrium distance req is the value of R for which e0 ðR; L; ZÞ  eat is minimized. The following theorem summarizes the situation for the
-model (numerical values are given to 4 decimal places): Theorem 2 (cf. ref. [2]). The energy curve e0 ðR; L; ZÞ  eat has: (i) a global strictly negative minimum if Z=L  0:3205; (ii) a local minimum (corresponding to a resonance of the molecule) if 0:3205 < Z=L < 0:4398 and (iii) does not have a local minimum if Z=L > 0:4398. To find the equilibrium distance, one computes that @R eðR; L; ZÞ ¼ R2 LZ ½GðxÞ  ZL1 , where GðxÞ :¼ x1 ð1 þ WÞ1 ðx þ WÞ2 W and where W ¼ Wðxex Þ. The function GðxÞ is found to be strictly increasing on the interval ½0; xG , where GðxG Þ ¼ 0:4398. Hence req ¼ G1 ðZ=LÞ for Z=L < 0:4398. The ground state energy of the molecule in the h
-model is emin ¼ eðreq ; L; ZÞ. Their asymptotic behavior as L ! 1 is given by   1 5 45 2 3 req ¼ 3=2 1=2 1 þ  þ  þ Oð Þ ; ð6Þ 4 32 2L Z   5 2 2 2 3 emin ¼ 2Z L 1  2 þ  þ Oð Þ ; ð7Þ 4 pffiffiffiffiffiffiffiffi where  ¼ Z=L. Although h
does not by itself provide numerically very good approximations for the ground-state energy and equilibrium distance of the real Hþ 2 -molecule for magnetic fields in the physically relevant range of

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3  109 –4  1013 Gauss, it can be used as the starting point of a perturbative calculation, as was done in ref. [1]. The equilibrium distance and the binding energy of the Hþ 2 molecule computed there were found to be in good agreement with earlier variational calculations. One consequence of these computations is the prediction of the existence, in fields B  1013 G, of He3þ 2 , a new atomic system, and a further example of the binding-enhancing properties of strong magnetic fields. 3 Equilibrium Nuclear Separation for H1 2 Using the arguments of ref. [5], Sect. 9, it can be shown that the ground state energy E0 ¼ E0 ðR; L; ZÞ of H can be estimated in terms of that of h
by jE0 ðR; L; ZÞ  e0 ðR; L; ZÞj  c
L; uniformly in R (recall that the constant c
of Theorem 1 is independent of R). One encounters a technical difficulty due to the existence of the second eigenvalue e1 of h
which, for large fields, becomes exponentially close to e0 , and prohibits a lower bound for the isolation distance of e0 of the type required for Theorem 1. However, all Hamiltonians under consideration commute with the z-parity operator Pz : z ! z, and if we decide right from the start to work in the ðPz ¼ 1Þ-eigenspace of even functions in z, e1 will not exist, and one can proceed as before. Using this estimate of jE0  e0 j < c
L and the fact that e0 ðL; Z; RÞ has a global minimum of order OðL2 Þ if Z=L  0:3205, one then shows, for sufficiently large B (and L), that the equilibrium distance Req of the true molecule (1) lies between the two roots R ¼ R1 and R ¼ R2 of the equation emin þ c
L ¼ eðR; L; ZÞ  c
L:

ð8Þ

A detailed analysis of this equation, using the known asymptotic large L-behavior of emin and of eðR; L; ZÞ ¼ x1 L2 Z 2 ðL1 Z  2x þ 4x2  10x3 þ Oðx4 ÞÞ, x ¼ RLZ, then shows that R1;2 ¼ ð2L3=2 Z 1=2 Þ1 þ OðL7=4 Þ (cf. ref. [2], Sect. 3). Hence: Theorem 3. For sufficiently large B, the ground state energy and equilibrium distance of the Hþ 2 -molecule (1) is given by E0 ðL; ZÞ ¼ 2L2 Z 2 þ 4Z 5=2 L3=2 þ OðLÞ;

ð9Þ

and Req ¼

1 2L3=2 Z 1=2

þ OðL7=4 Þ;

ð10Þ

respectively. 4 Discussion By Theorem 3, the internuclear distance tends to 0 as B ! 1. Despite the electrostatic repulsion between the two nuclei, a single electron suffices, under the influence of a strong magnetic field, to bring them arbitrarily close to each other. This is again an example of the binding-enhancing effect of strong magnetic fields. Furthermore, as B ! 1, emin ! 2Z 2 L2 , which is the ground state of 12 p2z  2ZL
ðzÞ, a

Non-Relativistic Hþ 2 -Molecule in a Strong Magnetic Field

137

one-dimensional Heþ -like ion with
-potentials. By ref. [5], Sect. 9, the ground state Heþ E0 ðBÞ of the true Heþ -ion in a strong magnetic field will lie within a distance of OðLÞ of 2Z 2 L2 . It follows therefore that Hþ

E0 2 ðBÞ ! 1; þ E0He ðBÞ

B ! 1:

ð11Þ

The conclusion is that as field strength increases, the Hþ 2 -model goes over into its united atom limit, the Heþ -ion. Several caveats are of course in order here. First of all, for values of B  4  1013 Gauss, for which the electron’s rest-mass becomes larger or equal than the lowest Landau level, our non-relativistic model should be replaced by a relativistic one (and ultimately of course nuclear effects will start to play a role). Next, the fixed-nuclei approximation is not realistic, and vibrational, and possibly also rotational, motions should be taken into consideration. Acknowledgement. This work has been supported by a CNRS (France)-Conicyt (Chile) collaborative grant. The work of (RaBe), (SPO), (RaBr) and (PV) have been supported by FONDECYT (Chile) project 102-0844, FONDECYT (Chile) project 302-0050, the EU Network ‘‘Analysis and Quantum’’,  (Czech) project 1763=2005, respectively. and FRVS

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