# A new vibrational level of the H molecular ion

###### Abstract

A new vibrational level of the H molecular ion with binding energy of a.u. neV below the first dissociation limit is predicted, using highly accurate numerical nonrelativistic quantum calculations, which go beyond the Born-Oppenheimer approximation. It is the first excited vibrational level of the 2p electronic state, antisymmetric with respect to the exchange of the two protons, with orbital angular momentum It manifests itself as a huge p-H scattering length of Bohr radii.

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31.15.-p###### pacs:

03.65.Nk###### pacs:

03.65.GeCalculations in atomic and molecular physics Scattering theory Solution of wave equations: bound states

The H molecular ion was one of the first non-trivial quantum mechanical systems treated in the early days of Quantum Mechanics [1]. Since, its study has been constantly pursued and the bound state calculations reach nowadays a very high degree of accuracy [2, 3, 4, 5, 6, 7].

The Born-Oppenheimer approximation provides us with a simple, although approximate, description of the system. Two electronic energy curves are correlated with the first dissociation limit: the 1s curve with a rather deep energy well supporting twenty vibrational states ( is the orbital angluar momentum) and the 2p which is mainly repulsive. The later one presents however at large distances a weakly attractive potential whose depth is about a.u. A single level () with binding energy a.u. has been found in this well up to now.

We will show in this letter that an excited level () of the 2p state of H actually exists. Its binding energy turns out to be extremely small – a.u., i.e. 30 neV or 0.00024 cm – and the corresponding wave function has a spatial extension of several hundred Bohr radii, a noticeable fraction of a micrometer. This state manifests itself in a huge p-H scattering length a.u. which dominates the low energy scattering cross section of proton by atomic hydrogen. The H formation rate, as well as the subsequent abundance of H molecules, can be substantially influenced by this resonant p-H cross section. The possible existence of an excited vibrational level was questioned in [3]

Calculations have been performed in the framework of non relativistic dynamics and in two different ways. On the one hand, by solving the Faddeev equations [8], what provides the low energy p-H scattering parameters and – by mean of the extended effective range expansion [9] – allows to determine the binding energies of near-threshold H states. On the other hand, by using adhoc bound state methods [7] which provide the best existing binding energies of the H system. Despite their radical differences, in their formal as well as in their practical implementations, these two approaches well agree in their bound state predictions.

We summarize in what follows the scattering and bound state results obtained by these two methods. As our calculations do not involve Born-Oppenheimer approximation, we use only exact quantum numbers in non-relativistic dynamics, where spin-orbit and spin-spin couplings are neglected (their effects are discussed below) i.e.: the total orbital angular momentum , the parity and the symmetry property of the spatial wave function with respect to the two-protons exchange; for the latter symmetry, the symmetric states have a total proton spin (singlet) and the antisymmetric ones have (triplet). The connection between exact and approximate quantum numbers is discussed in [7]: for the 1s levels are singlet, even parity states, while the 2p levels are triplet, even parity states.

The 3-body (p,p,e) Faddeev calculations are performed in configuration space [8]. Three sets of Jacobi coordinates, corresponding to the different asymptotic states, are involved and defined by

where denote cyclic permutations of (123), the particle masses and we identify 1p, 2p, 3e. The standard Faddeev equations read

(1) |

where is the 3-particle free hamiltonian and the 2-body Coulomb potential for the interacting pair.

Equations (1) provide satisfactory solutions for bound states but are not suitable for scattering Coulomb problems. The reason is that their right hand side does not decrease fast enough to ensure the decoupling of the Faddeev amplitudes in the asymptotic region and to allow unambiguous implementation of boundary conditions. In order to circumvent this problem, Merkuriev [10] proposed to split the Coulomb potential into two parts by means of some arbitrary cut-off function

and to keep in the right hand side of equation (1) only the short range contribution. One is then left with a system of equivalent equations

(2) |

in which is a 3-body potential containing the long range parts:

The systems of equations (1) and (2) are strictly equivalent to the Schrödinger equation and realize two different partitions of the total three-body wave function

This approach was found to be very efficient and accurate in calculating the low energy ePs and eH cross sections [11].

Equations (2) were solved by expanding amplitudes in the bipolar harmonics basis

(3) |

and their reduced components in the basis of two-dimensional splines. Some care has to be taken in extracting the scattering observables, specially at zero energy, from the asymptotic solution at finite distance. The long range polarization force – generated by the equations themselves – makes the convergence of the observables as a function of the p-H distance very slow and requires an appropriate extrapolation procedure. The first results of these calculations can be found in [8] and a detailed explanation of the method will be given in a forthcoming publication [12].

For the 1s state, we obtain the scattering length a.u. The zero energy Faddeev amplitude has 20 nodes in -coordinate, indicating the existence of 20 vibrational energy levels for H.

In the 2p state, our calculations gives a.u. The sign and the nodal structure of the Faddeev amplitudes indicates that such a big value is due to the existence of a first excited state with extremely small binding energy. By calculating the p-H phase shifts at small energies and using the effective range theory we are able to determine its binding energy. In presence of long range p-H polarization potential, the effective range expansion has the form [9]

(4) |

where is the wave vector and are coefficients depending on the interaction. It turns out that, in the presence of a weakly bound state with imaginary momentum , the and terms in (4) become negligible and the expansion recovers the standard form [9, 13]:

(5) |

with . Coefficients and are obtained by fitting the phase-shift and they determine the value. Using , we found [8] by this procedure a bound state at a.u. below the first p+H dissociation threshold. To our knowledge, this is the weakest bond ever predicted, three times smaller than the He atomic dimer [14].

The S-wave p-H cross section for the triplet () state is displayed in fig. 1 (dotted values). The singlet () contribution is negligible in the zero energy region. It is interesting to compare the 3-body calculations with those (solid line) provided by the simple Landau’s two-body potential [15]. This model – based on Pauli repulsion between protons overbalanced at 10 a.u. by attractive polarization forces – gives quite a good result for the ground state a.u. instead of the exact value [7]) but differs strongly in the predictions of the first excited one a.u.). In the zero energy scattering, both calculations differ also by more than one order of magnitude while at energies 10 they are already in quite a good agreement.

The positive sign of the p-H scattering length and the existence of one node in the Faddeev amplitude (see fig. 2) unambiguously shows the existence of an excited bound state. After this calculation was done, we decided to use a direct method in order to obtain a more accurate value of the binding energy as well as a direct computation of its wavefunction. Because the three particles are bound, the wavefunction must decrease exponentially if any of the three inter-particles distances goes to infinity. The idea is thus to expand the full 3-body wavefunction on a convenient discrete basis set and to diagonalize the 3-body Hamiltonian in this basis set. For highly accurate calculations of very weakly bound states, the basis set must be chosen carefully. The first step is to isolate the angular dependance of the 3body wavefunction, which is straightforward for states. One is left with a 3-dimensional Schrödinger equation depending on the inter-particle distances only. We use the perimetric coordinates

(6) |

and express the Schrödinger equation as a generalized eigenvalue problem for the energy :

(7) |

where and operators are polynomials in the operators. The basis functions used in the calculation are direct products of Laguerre polynomials and exponentials along each perimetric coordinate, whose properties are discussed in details in [7]. The matrices representing and in such a basis set are real symmetric sparse banded matrices, where all matrix elements are known in analytic form and involve only simple algebraic expressions. The generalized eigenvalue problem is then solved using the Lanczos algorithm in order to produce few eigenvalues, among several thousand, in the interesting energy range.

Whereas an accurate computation of the ground level of the 2p state requires only a moderately large basis set (about 20,000), the computation of the first excited state is much more difficult and requires at least a basis size of 150,000 and a careful choice of the variational parameters of the basis. We used basis sizes up to 450,000 to confirm that the results discussed here are fully converged. For the first excited vibrational level we find a total energy a.u. and a dissociation energy a.u. This gives a binding energy a.u. with an uncertainty of the order of a.u., due to numerical precision rather than to the basis size. This value is consistent with the one obtained by the scattering method, but more accurate.

We have also calculated the wavefunctions. These are full three-body wavefunctions and are thus not easily plotted. However, they take significant values only for rather large internuclear distances: the electronic wavefunction is thus essentially the ground state of the hydrogen atom attached to one of the protons, independently of the internuclear distance. We checked that this simple property is almost exactly obeyed by the three-body wavefunction. Once this trivial part of the wavefunction is factored out, one is left with a wavefunction depending only on the internuclear distance. It is plotted in fig. 2 for the lowest two levels. Because of the very large size of the excited state, we chose to plot the wavefunction using a logarithmic scale for the internuclear distance The ground level is a nodeless wavefunction centered around a.u., while the excited level extends much further; it has a maximum at a.u. and still significant values at a.u. There is an inflexion point at a.u., located at the outer turning point of the Born-Oppenheimer (or Landau) potential, where the 2p potential equals the binding energy. In the same figure, we also plot the zero-energy wavefunction got from the Faddeev approach. At small it is remarkably similar to the wavefunction of the excited level, which is not surprising considering the very small energy difference. At large distance, the zero-energy wavefunction diverges linearly and has a zero at , the scattering length value 750 a.u.

Results presented above have been obtained with the mass ratio , as recommended in [16]. Using the more recent 1998 CODATA value [17] would not change anything significantly. Our method makes possible to compute the energy levels for any mass ratio of the particles. When the mass ratio is decreased, the binding energy also decreases until a critical value beyond which the excited level disappears and one is left with a single bound state. We estimate this critial mass ratio to be around 1781. This is rather close to the actual value which explains why the state is so weakly bound. The closeness of the critical mass ratio also explains why the Landau potential discussed above – although it is fairly accurate – gives a wrong binding energy.

Calculations presented here were performed using a fully non relativistic dynamics with just Coulomb pair-wise interactions taken into account. In view of the extreme sensitivity of these results, it is necessary to quantify the possible relativistic effects.

As the dynamics is governed by a shallow potential well at distances around 10 a.u., where the nuclear motion is very slow, relativistic corrections must be considered only for the electronic motion. As the internuclear distance is much larger than the typical electron-nucleus distance, relativistic corrections will be essentially identical for the weakly bound state and the dissociation limit, leaving the binding energy only very weakly modified. The first order relativistic and radiative corrections for the H states have been obtained in [18] and are discussed with some detail in [3]. The results summarizing the relative corrections to the binding energy () for the 1s (filled circles) and 2p (filled square) states are given in fig. 3. For 1s states they are smaller than and vary smoothly over five decades of binding energy. For the 2p level they are one order of magnitude smaller. In absolute value they are a.u. and as they scale as [18], they should be at least one or two orders of magnitude smaller for . Thus, despite the smallness of its binding energy, relativistic effects preserve the bound character of the new state we have presented.

Another source of relativistic corrections, not included in Refs. [18, 3], is the modification of the polarization potential at very large distances due to retardation effects (Casimir-Polder effect). Contrary to the dipole-dipole case (Van der Waals forces), where the long-range behaviour is changed into , in the charge-dipole case we are considering the usual term is modified by adding a contribution [19] in the form:

(8) |

where is the fine structure constant. At a.u., the correction turns out to be negligible.

One should be careful with the spin-orbit and spin-spin interactions. As all states considered here have zero total angular momentum , the spin-orbit coupling vanishes and all states correspond to (where is the sum of the total angular momentum and the electronic spin The spin-spin interaction is more tricky. Indeed, we are interested in triplet states with total two-proton spin , so that the total angular momentum can be either or This hyperfine structure should be very close to the hyperfine structure of the hydrogen atom. As the latter is much larger than the binding energy we have calculated for the non-relativistic problem, it is likely that the level lies above the dissociation limit of the series. The dissociation rate induced by the hyperfine coupling is however most probably very low.

A direct measurement of the p-H cross section at very low energy seems unlikely at present. One can however access the low energy p-H continuum in the final state of the H photodissociation cross section. The excited vibrational 2p level predicted here is radiatively coupled to the 1s level. The electric dipole transition between those two levels should be observable in the 6 GHz range using an experiment similar to the one used to detect the transition [20, 21]. An experimental confirmation of our results would be very interesting.

###### Acknowledgements.

The authors are sincerely grateful to C. Gignoux, N. Billy and B. Grémaud for useful discussions and helpful advices. The numerical calculations were performed at CGCV (CEA Grenoble) and IDRIS (CNRS). We thank the staff members of these organizations for their constant support. Laboratoire Kastler-Brossel de l’Université Pierre et Marie Curie et de l’Ecole Normale Supérieure is UMR 8552 du CNRS.## References

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