## Abstract

We discuss the Jahn–Teller mechanism for dissociative recombination in low energy collisions between electrons and ions, in energy ranges relevant to the processes underway in interstellar clouds. While theory has become capable of predicting recombination rates in reasonable agreement with storage ring experiments, some discrepancies remain with them, and a long-standing discrepancy with stationary afterglow measurements remains troubling. Speculations about the desirable improvements in both theory and experiment are presented.

## 1. Introduction

The simplest polyatomic molecular ion, , serves as a prototype for electron-induced bond-breaking collisions. Because the basic theory of dissociative recombination (DR) was established over 50 years ago by Bates (1950), one might think that both the fundamental mechanisms operating and the quantitative theoretical description for this simple target molecule would have been well understood by the year 2000. In fact, theory struggled to understand the DR process for this simple system at low energies. This was brought into sharp focus by Bates *et al*. (1993), 43 years after Bates' important 1950 paper, when those authors labelled the DR problem ‘enigmatic’. This label was earned because the DR rate had been measured in most experiments to be significantly higher in the sub-electronvolt energy range than was imaginable, according to early theoretical estimates based on the tools then available. For a comprehensive review of the experimental situation up until a few years ago, we steer the reader to an excellent paper by Larsson (2000), which traces the rich history including more than 30 publications cited that deal with laboratory observations.

The conventional recombination process typically occurs (O'Malley 1966) when a Born–Oppenheimer potential surface for the (ion plus electron) neutral complex crosses through the ionic potential surface in the vicinity of the ionic ground state. At high incident electron energies, around 9 eV in fact, direct electron capture into a repulsive, doubly excited state occurs and it results in a high recombination rate for . This has been observed experimentally by Larsson *et al*. (1993) and confirmed in the theoretical calculations of Orel & Kulander (1993). Both theory and experiment obtain peak DR cross-sections in this energy range in the vicinity of 10^{−16} cm^{2} and a rate coefficient around 10^{−8} cm^{3} s^{−1}. Even though this theoretical calculation was carried out for the simplified ‘T-shaped’ geometry only, it successfully accounted for the direct resonant part of the DR process for hot electrons.

In most astrophysical environments where is important, however, like the interior of an interstellar cloud, the relevant energy range where recombination must be understood is centred at much lower electron energies, 2–100 meV. Early theoretical musings about made it difficult to rationalize the high DR rate observed in early experiments, in the absence of any plausible direct route for the process. Alternative ‘indirect’ mechanisms for DR have been identified in the past (Bardsley 1968; Giusti 1980; Nakashima *et al*. 1987; Schneider *et al*. 2000). The most important are those for which the incident electron is captured into an autoionizing Rydberg state, after which the molecule vibrates with enhanced amplitude, sufficiently long to redirect flux into a dissociative pathway. A first exploration (Schneider *et al*. 2000) of such pathways for confirmed that at low energies, significantly higher rates could result than would be possible from the direct pathways alone, but the rough estimates possible at that time still seemed 2–3 orders of magnitude lower than the rate measured in storage ring experiments.

A step forward taken in a series of papers was the recognition that Jahn–Teller distortion probably plays an important role in the recombination process. An early estimate of this mechanism (Kokoouline *et al*. 2001) gave encouraging signs, but the calculated DR rate was still around an order of magnitude lower than that measured in the storage ring experiments. However, a subsequent, more detailed study (Kokoouline & Greene 2003*a*,*b*, 2004*a*) observed that the 2001 results had been based on a misinterpreted reaction matrix convention dating back to the spectroscopic studies of Stephens & Greene (1994, 1995). When this correction was introduced, the calculated DR rates increased by approximately a factor of *π*^{2}, giving a low-energy rate comparable to the best new experimental measurements (McCall *et al*. 2003, 2004; Kreckel *et al*. 2005) using a much colder ion source than previous storage ring experiments could achieve. Some discrepancies that existed in 2003 disappeared once additional factors were taken into account in the experiment–theory comparison, including a more careful convolution over the storage ring energy spread and detailed modelling of the so-called ‘toroidal correction’. Encouraging agreement was also achieved with some of the energy ranges studied by Helm and co-workers (Bordas *et al*. 1991; Mistrik *et al*. 2000) in their clever experiments on H_{3} Rydberg state photoionization, which gave further confidence in the new theoretical framework.

Nevertheless, a number of problems still remain to be understood in this challenging system. The present paper reviews the progress that has occurred over the past 5 years, but with a critical eye turned towards identifying the theoretical and experimental difficulties that remain to be solved.

## 2. H_{3} Rydberg states and recombination

Molecular Rydberg states challenge our conventional understanding and intuition, which is based primarily on the Born–Oppenheimer approximation. In valence electronic states, the electronic velocities are so fast compared to the nuclear vibrations and rotations that it makes sense to treat the nuclear motion adiabatically. As electrons are excited to higher and higher Rydberg states, their motion eventually slows to the extent that the Born–Oppenheimer concept of adiabatic nuclear motion becomes a poor approximation. Yet for penetrating, low orbital angular momentum states of an electron, part of its journey is spent within the short-range core region where the electron moves rapidly owing to its conversion of Coulombic potential energy into kinetic energy. Thus, a variant of the Born–Oppenheimer approximation still applies; only its validity is restricted to this short-range portion of the electron motion. An effective theoretical framework to treat molecular Rydberg states grew out of the ideas of Chase (1956, 1957), Arthurs & Dalgarno (1960), Mulliken (1966), Fano (1970), Jungen & Dill (1980) and Seaton (1983). The modern version of this framework is today called multichannel quantum-defect theory (MQDT; Matzkin *et al*. 2000; Hamilton & Greene 2002).

This class of methods has proven capable of describing the interconversion between energy in electronic form and in the vibrational or even dissociative degrees of freedom. Remarkably, the theory is able to incorporate an important class of non-adiabatic phenomena in molecules such as H_{2}, even though the starting input information is primarily the quantum-defect function *μ*_{lΛ}(*R*) or its generalizations into the form of a quantum-defect coupling matrix. Specifically, it is the *R*-dependence of the quantum-defect function—or, more generally, the electron–molecule scattering matrix—that enables the interconversion of energy between electronic and vibrational (or dissociative, bond-breaking) forms.

Experimental observations of H_{3} Rydberg spectra have been performed by more than one Nobel laureate, beginning with G. Herzberg (1979) and followed later by W. Ketterle and co-workers (Dodhy *et al*. 1988). To analyse some aspects of the spectra, Fano & Lu (1984) stressed the importance of this problem, and it led to an application of MQDT ideas to treat rotational *l*-uncoupling of p-states of the H_{3} molecule by Pan & Lu (1988). Further efforts by Bordas *et al*. (1991) developed important theoretical generalizations as part of their analysis of a series of beautiful experiments on H_{3}, including more recent studies of three-body predissociation (Galster *et al*. 2004). An important generalization of MQDT to handle vibrational channel interactions mediated by the Jahn–Teller effect in H_{3} was proposed by Staib & Domcke (1990), with extensions and applications by Stephens & Greene (1994, 1995). These last three papers mentioned gave evidence that the most rapid vibrational autoionization pathways for *n*p-Rydberg states of H_{3} are, in fact, controlled by Jahn–Teller distortions, but the form of molecular MQDT adopted at that time made it challenging to see how the DR pathway could be included in the theoretical description. For diatomics, Jungen and collaborators formulated a number of different ways to incorporate dissociation dynamics (Jungen 1984; Greene & Jungen 1985; Gao *et al*. 1993), culminating in an effective method (Jungen & Ross 1997) that has been applied with great success to H_{2}.

Generalization of the treatment of Jungen & Ross (1997) to handle polyatomic dissociation, where the nuclei can move in multiple dimensions, appears to be less than straightforward. For this reason, one key step towards a treatment of the dissociative channels H_{2}+H and H+H+H, which arise in the system, was the recognition that in the hyperspherical coordinate system, there is only *one* dissociative coordinate: the hyper-radius *R*. Instead of studying the adiabatic potential energy curves of the ionic target molecule ground state and the neutral molecule resonant states as functions of the internuclear distance in a diatomic, we plotted (figure 1) these adiabatic energies as functions of the triatomic ion's nuclear hyper-radius (Kokoouline *et al*. 2001). With this modification, it is possible to immediately apply the techniques from established DR theory for diatomics. One reason why this is helpful, conceptually, is that for any Born–Oppenheimer potential surface for a triatomic, there will be an infinite number of adiabatic hyperspherical potential curves extending all the way to positive infinity. Moreover, any neutral hyperspherical potential curve *U*_{i}(*R*) lying higher than the ionic ground-state energy will have an autoionization width *Γ*_{i}(*R*) that can be used in the O'Malley (1966) formula for the DR cross-section,(2.1)Here, *Γ*_{i}(*R*_{i}) is the partial fixed-hyper-radius ionization width in the incident channel, evaluated at the Condon point *R*_{i}; *E*, the incident electron energy; and , the slope of the *i*th excited-state neutral hyperspherical potential curve that can capture the incident electron. *Ψ*(*R*_{i}) represents the hyper-radial vibrational wave function of the initial ionic target level, evaluated at the Condon point for the incident energy of interest. Finally, *s*_{i}(*E*) is the survival probability; it is often approximated by unity, for systems like H_{3} where it is comparatively unlikely that the system will autoionize before finding its way to dissociation, following electron capture. These pathways are shown as functions of the hyper-radius in figure 1.

This technique of mapping the multidimensional vibrational motion onto a single dissociative coordinate should be useful in other polyatomic systems, but it should be kept in mind that the adiabatic hyperspherical approximation works better for some systems than for others. For instance, we have found that it works quite well for symmetric systems like and (Kokoouline & Greene 2003*a*,*b*), but requires improvement for the asymmetric isotopologues H_{2}D^{+} and D_{2}H^{+}. Accordingly, our treatment of the asymmetric target ions (Kokoouline & Greene 2004*b*, 2005*a*) explicitly includes non-adiabatic coupling among the hyperspherical potential curves of the ion.

We stress that a number of essential ingredients of our calculations were provided by other researchers, beginning with accurate Born–Oppenheimer potential energy surfaces for the triatomic ion (Cencek *et al*. 1998; Jaquet *et al*. 1998). Next, a number of potential energy surfaces are needed for the *n*p-Rydberg states of neutral . Here again, we had the good fortune that these were already determined by M. Jungen in extensive *ab initio* quantum chemistry calculations carried out in the course of a joint theoretical and experimental paper (Mistrik *et al*. 2000) on photoionization. These potential energy surfaces of the Rydberg states are nearly ideal for an MQDT analysis, because their quantum-defect functions display only a weak dependence on the principal quantum number *n*. On the other hand, the *D*_{3h} symmetry of this equilateral triangle ion implies the existence of a conical intersection at the symmetry point, with its usual problematic divergence of the non-Born–Oppenheimer coupling terms. These are the reasons why a diabatic parameterization of the surfaces in terms of a single 2×2 quantum-defect matrix for the in-plane p-orbitals and a 1×1 quantum defect for the out-of-plane p-orbital, in the spirit of the analysis by Staib & Domcke (1990), turns out to be so effective for this problem. Yet another key aspect of the calculation, from which we benefitted tremendously, was the existence of a robust method for calculating the adiabatic hyperspherical potential curves (Macek 1968; Lin 1995) of the triatomic ion. This robust basis-spline method had been developed by Esry into an efficient and general program that had originally been aimed at understanding Efimov states and three-body recombination of bosonic atoms at ultracold temperatures (Esry *et al*. 1996, 1999; Esry 1997).

Two different levels of theoretical sophistication were applied at this point to manipulate the set of aforementioned data (the potential surfaces of the ionic and Rydberg p-state molecules, the ionic hyperspherical potential curves and the diabatic quantum-defect model of the Rydberg p-states that dominate recombination) into predictions of the DR rate. The first or ‘simplified’ model is an adaptation of the famous direct DR formula of O'Malley (1966), as it treats the electronic and vibrational degrees of freedom quantum mechanically, but without quantizing the final vibrational energy levels in the hyper-radial potential curves. In other words, equation (2.1) is applied after first using MQDT at a *fixed hyper-radius* to map out the real and imaginary parts of the resonance potential curves *U*_{i}(*R*). This simplified model is comparatively rapid to implement because omission of the hyper-radial quantization of the neutral molecule resonance states results in an automatic averaging over the dense forest of individual resonances that, in fact, characterize the exact spectrum.

Note that only capture into the 2p channels depicted in figure 1 can lead immediately to direct dissociation of the system. However, we find that the DR rate contributed by these direct channels is at least 10-fold smaller than via indirect Rydberg channels, such as the 3p pathways shown. Recombination via the indirect pathways is initiated in this picture by capture into one of the 3p potential curves, after which the system evolves away from the Franck–Condon region of the ionic ground state. After evolving to larger hyper-radii, though, the system rebounds because it eventually encounters a classical turning point and is reflected. It thus cuts back through the forest of dissociative hyperspherical potential curves again and again, with each such reflection, and eventually makes a non-adiabatic transition onto the dissociative surface, producing either H_{2}+H fragments or else H+H+H. Our calculations suggest that indirect dissociations that occur following captures into principal quantum numbers around *n*≈4–5 dominate the recombination rate.

The second or ‘advanced’ description is more sophisticated and also more demanding computationally. It treats every degree of freedom in the problem quantum mechanically, at least in some level of approximation, whereby a dense forest of recombination resonances emerges that must be averaged over to connect with any experiment conducted at finite energy resolution. DR of is basically a version of the four-body problem, since two electrons in the triatomic ion are effectively frozen into the ground-state orbitals and can be viewed as frozen out of the problem, leaving the incident electron plus three nuclei to be treated. The electron dynamics in the field of vibrating, rotating, nuclei is handled by a rovibrational frame transformation (Kokoouline & Greene 2003*b*). Vibrations of the nuclei in their centre of mass body frame, on the surface, constitute a three-dimensional non-separable problem in Schrödinger wave mechanics and the quantum rotations involve three Euler angles, but can be handled using closed-form Wigner *D*-matrices in a rigid rotor approximation. Altogether, our approach can be viewed as a ‘divide-and-conquer’ strategy applied to the nine-dimensional Hilbert space of this four-body problem. Note that in order to carry out an accurate energy convolution to determine the average DR rate in any energy range of interest, it is crucial to trace out the full resonance linewidth even for narrow, but very high, resonances. This has been accomplished in our study by solving the final set of (linear) MQDT equations on a fine energy mesh with hundreds of thousands of energy mesh points per symmetry over the range 0–2 eV. This is the most laborious numerical part of the calculation, as it requires the solution of a linear system of the order of 4000×4000 at each such energy, and at each of about two dozen symmetries.

The results of the simple and advanced treatments are shown in figure 2 along with the Jensen *et al*. (2001) storage ring experiment. Both of these models account reasonably well for the average low-energy DR rate, but one sees that even the coarse-grained version of the advanced description exhibits modulations associated with resonance physics. For many purposes, such modulations may not be of interest, because the thermally averaged DR rate is often the quantity of interest; for such applications, our simpler treatment that sidesteps the step of quantizing the ionic rotations and hyper-radial vibrations should be viewed as an attractive theoretical approach. But for the triatomic hydrogen ion, the simplest polyatomic, it remains desirable to explore in the greatest detail possible the full high-resolution spectrum, both with an eye towards improving theoretical methods and also to help identify potential systematic errors that might plague experiments.

Figure 3 shows the prevalence of sharp resonances clearly in the higher-resolution convolution of the advanced treatment. The comparison of our quantitative theoretical description with the first storage ring experiment to be carried out using rotationally cold target ions (McCall *et al*. 2003) shows generally good agreement between theory and experiment. Moreover, this agreement applies not only to the general order of magnitude of the overall rate, but also in the similarities concerning many individual resonances, shoulders and sharp decreases with energy.

While there is reason for optimism in the general agreement between theory and experiment, some clear discrepancies are evident in figure 3. To see them more clearly, we plot the same comparison over a smaller energy range on a linear–linear scale in figure 4, which makes more apparent the lower theoretical DR rate over the energy range 0.04 up to about 0.12 eV. The theoretical resonance modulations are evidently stronger than that shown by the experimental DR spectrum. A number of possible interpretations of this discrepancy come to mind. This could simply represent a limitation of one or more of the theoretical approximations that have been employed in the calculation, which are as follows.

One of the most severe is the use of the adiabatic hyperspherical approximation to calculate the vibrational wave functions of the triatomic ion. This approximation was improved in the subsequent studies of H

_{2}D^{+}and D_{2}H^{+}recombination (Kokoouline & Greene 2005*a*), by inclusion of non-adiabatic coupling among the adiabatic hyperspherical potential curves. Incorporation of that coupling reduced the typical error in the lower vibrational energy levels from the range 50–200 cm^{−1}, in the adiabatic approximation, to only 1–2 cm^{−1}when non-adiabatic coupling is included.A second approximation that might be improved in the future is the use of a symmetric top rotational wave function to describe the higher angular momentum states of . It is not overly difficult to calculate these energies separately for each ionic angular momentum of interest, but this was not implemented in our first theoretical efforts in an attempt to improve the economy and simplicity of the calculation.

A third approximation is the use of a diabatic Jahn–Teller parameterization of the fixed-nuclei

*K*-matrix, which was fitted (Mistrik*et al*. 2000) to the*ab initio*Rydberg surfaces over a limited range of internuclear distances. If more global fit is carried out, it should provide a better description of the vibrational dynamics at larger internuclear distances. This may, in fact, be important to accurately describe the branching ratios among the various DR products. The body-frame energy dependence of the fixed-nuclei*K*-matrix has also been neglected in our calculations to date, and this is another approximation that might be improved, although the quantum defects seem to have a comparatively weak energy dependence for the p-states of H_{3}.The theoretical framework of our entire treatment has been based on the assumption that the relevant electron partial wave is a ‘p-wave’. This is only approximately true, of course. In reality, if a p-wave electron collides with the ion, it will scatter with some probability into other partial waves, primarily s- or d-waves. However, test calculations (S. Tonzani 2005, unpublished work) suggest that the probability of p-wave scattering into other partial waves is small, at a few per cent level.

While one is accustomed to assuming that any discrepancy between theory and experiment in such complicated systems must derive from imperfections in the theoretical description, one should keep in mind that this has not been firmly established in this case. The probable explanation of the disagreements may be the theoretical approximations employed, but one should not rule out possible experimental issues as well, which are as follows.

Some fraction of the target ions in the storage ring may be rotationally or (less probably) vibrationally excited. This tends to smear out resonance structures that were calculated theoretically for a single rovibrational quantum state of the target ion. In the actual storage ring experiments, excitations could be induced by either collisions with background gas or the toroidal regions where electrons are bent into the path of the circulating ion beam. In fact, there are electron collisions in those toroidal regions that are at several electron volts of relative energy, although estimates carried out for this effect concluded that only a small fraction of the ions (less than 10%) should be in excited states. A cautionary note about those estimates worth mentioning is that they were based on the rotationally inelastic electron scattering cross-sections computed by Faure & Tennyson (2002), which we believe (Kokoouline & Greene 2003

*b*; see esp. table VI) may be significantly underestimated at low energies. Even if the inelastic electron scattering cross-sections are larger than that computed by Faure & Tennyson (2002), this does not necessarily mean that a larger fraction of the ions will be excited, since this means that the rates of de-excitation will be enhanced as well as the rates of excitation. The main point of this paragraph is to suggest that a more complete analysis of this aspect of the storage ring experiments may be warranted, to help understand the state of the ions in the ring more quantitatively. Ideally, it would be best if one could perform spectroscopic measurements directly on the ions while they are inside the storage ring, but such measurements do not appear to be feasible in the near future.One of the significant remaining puzzles is still the matter of why some afterglow experiments (e.g. Plasîl

*et al*. 2002) see such a strong dependence of the ‘effective DR rate’ of on the density of diatomic hydrogen molecules in the interaction region, and why the low H_{2}density limit of this effective DR rate is very low compared to theory and the storage ring experiments. Some preliminary explorations (Mikhaylov & Kokoouline 2005, unpublished work) suggest that the observed dependence on H_{2}density could result if there is some mechanism for producing H_{3}metastable Rydberg molecules in the cell. Moreover, the gas-phase two-body recombination rate is close to the limiting apparent rate measured at*high*H_{2}density instead of at the expected low density limit. Further examination of this possibility remains desirable, to see whether it will produce an understanding of why these very different experimental methods appear to give such different apparent DR rates for at low energies.

Calculations have also been carried out (Kokoouline & Greene 2005*a*) for two isotopologues, D_{2}H^{+} and H_{2}D^{+}, having *C*_{2v} symmetry, which may help to assess the robustness of the theoretical approach developed for DR. Figures 5 and 6 show our resulting theoretical calculations of the DR rate, compared with some of the available measurements. Of course, the experimental measurements have not been carried out as exhaustively as has been studied; this is reflected by the poorer resolution evident in the data shown in figures 5 and 6. The agreement between theory and experiment can be immediately seen to be distinctly poorer than the comparisons that were obtained previously for and (Kokoouline & Greene 2003*b*). The origin of these discrepancies is still not understood. The upshot is that despite the strides taken in our theoretical understanding of the DR process for this most fundamental of all polyatomic molecules, theory still needs significant improvement and extension if it is to achieve reliable, quantitative predictive power of the recombination rate and other observables.

## Discussion

D. Zajfman (*MPI and the Weizmann Institute*). It seems that your calculated cross-sections for dissociative recombination of always have a more structured landscape than the experiments. Do you have any hints why?

C. H. Greene. Our first calculations were significantly more structured than the experiments, which Andreas Wolf helped us trace to a misconception on our part. We had originally thought that the convolution over the 0.1 meV spread in the parallel energies was unnecessary because it seemed so close to a Dirac delta function. However, Andreas explained how even this small spread plays a key role in smoothing over the sharp resonances, especially as the energy increases. Once we incorporated this correction, the discrepancies between the sharpness of theoretical and experimental resonances became much smaller. However, there does still appear to be a somewhat sharper structure in the theoretical resonances than in the experimental ones. We have no reason to suspect that theory should predict resonances that are too narrow, although it is of course a possibility. It also seems conceivable that more rotational levels are populated in the stored ion beam than has been estimated for these experimental conditions, owing to rotational excitations that can occur each time the ions circulate through the toroidal region.

S. Miller (*University College, London*). You show values for dissociative recombination as a function of H_{2} number density. They show an increase of approximately 50 when *N*(H_{2}) increases from approximately 10^{11} to 10^{12} cm^{−3}. This is a very important density region for in planetary atmospheres. How realistic are these variations and over what temperature range are they applicable? What is the effect of temperature on these numbers?

V. Kokoouline. In the afterglow experiments done by Glosik's group, the dissociative recombination rate *α* depends strongly on the density *n* of present H_{2} molecule. (In their experiments, ions are always surrounded by H_{2}.) A similar behaviour is observed for the D_{3}^{+} isotopologue by Glosik. This result contradicts with a number of recent experiments done in storage rings. One possibility to explain the dependence *α*(*n*) is to assume presence of long-living states of that are produced in Glosik's experiment during the plasma discharge or later on. Our simple model shows that such assumption would give the observed *α*(*n*). Another important assumption in our model is that such long-lived states are destroyed due to the collisions with H_{2}. The possible candidates for such states are excited Rydberg states *n*p*σ* and *n*d states. The question whether or not such states can be present in Jupiter's atmosphere is not clear.

H. Helm (*University of Freiburg, Freiburg, Germany*). What is the dominant origin of the width of your dissociative recombination resonances? Is it inverse autoionization or predissociation which is responsible?

C. H. Greene. We have not carried out a systematic study of this question, but one piece of evidence from the experiments and from the theory appears to provide an important clue. The dissociative recombination rate drops precipitously once the energy rises above the first excited vibrational thresholds, 01^{1}E at 0.31 eV, 10^{0}A_{1} at 0.39 eV. Then there is a second precipitous drop when the energy is raised above the next two vibrational thresholds, 02^{2}E and 11^{1}E at 0.62 and 0.69 eV, respectively. This is indirect evidence that while resonance dissociation can compete effectively with autoionization for energies less than 0.3 eV above the ground state, the balance tips strongly towards autoionization as the dominant decay channel above this energy once a new vibrational autoionization channel becomes energetically allowed.

A. Orel (*University of California, CA, USA*). As you raise the electron temperature, it is possible to have a mechanism similar to H_{2} where there is capture into a Rydberg state, and then coupling to the approximately 9.5 eV electronic resonances. Has this effect been considered?

C. H. Greene. The direct capture into that channel will not of course be relevant until energies much higher than we have considered. At low energies, we have not tracked exactly where the dissociative flux evolves at distances more than a few Bohr radii beyond the equilibrium. Our impression is that if the nuclei can reach such large hyper-radii as would be necessary to reach this channel you're alluding to, dissociation is already highly probable even without the presence of that electronically excited surface. For this reason, one might expect that this channel (whose effects are entirely omitted from our calculations) will not dramatically change the total recombination rate when it opens up, although it will almost certainly introduce a new dissociative pathway that will modify the dissociative branching ratios.

B. McCall (*University of Illinois, Urbana, IL, USA*). Do you have any ideas for what might be the source of the discrepancy between your calculations and the storage ring measurements just short of 0.1 eV (aside from the possibility that the experiment is wrong)?

C. H. Greene. There are some limitations to the approximations we have made, and over time we will explore them to see how sensitive our final results will be to successive improvements. One such approximation is the use of the adiabatic hyperspherical approximation for the vibrational wave functions of the target ion. We know how to improve this approximation and in fact we have done so already for the isotopologues H_{2}D^{+} and D_{2}H^{+}, and hopefully this will be implemented in the near future for too. A second improvement that could be made is the incorporation of the energy dependence of the body-frame quantum-defect function for p-waves, which is the starting point of our approach. Yet another improvement would be to develop a full treatment of orbital angular momentum mixing, which appears to be small but which is expected to arise at some level.

## Acknowledgments

This work has been supported by the National Science Foundation under grant nos PHY-0427460 and PHY-0427376, and in some earlier stages by the Department of Energy and National Energy Research Scientific Computing Center (www.nersc.gov). We thank B. Esry for his key participation in the early stages of this project, and for providing his unpublished computer programs to calculate hyperspherical potential curves. Discussions with A. Larson, S. Tonzani, I. Mikhaylov, A. Wolf, B. McCall, H. Kreckel, D. Zajfman, M. Larsson and T. Oka have been greatly appreciated.

## Footnotes

One contribution of 26 to a Discussion Meeting Issue ‘Physics, chemistry and astronomy of ’.

- © 2006 The Royal Society