## Abstract

The dissociative recombination (DR) of H ions with electrons, producing neutral atomic and molecular fragments, is driven primarily by the vibronic Jahn–Teller (JT) interaction between the electronic components of the *pe*′ e^{−}–H collision (Rydberg) channel. The JT parameters characterizing this interaction are therefore of great interest as they are required for the theoretical predictions of the DR cross section. In this contribution, we review various determinations of these quantities that have been made previously, based both on spectroscopic studies of 3*pe*′ Rydberg-excited H_{3} states, and on the analysis of the corresponding *ab initio* H_{3} Rydberg potential surfaces near the conical intersection (D_{3h} symmetry) for *n*=3−5. The highly correlated theoretical 3*pe*′ potential surfaces of Mistrík *et al.* are used for a new determination of both the linear and quadratic JT terms.

## 1. Introduction

Ever since it was recognized by Kokoouline *et al*. [1] that the electron capture process in the dissociative recombination (DR) of H is driven by the vibronic Jahn–Teller (JT) interaction between the degenerate components of the *pe*′ Rydberg channel of the H_{3} molecule, increasingly sophisticated theoretical evaluations of the DR rate coefficient have been made that were based on this hypothesis. These computations have culminated in the work of dos Santos *et al.* [2], which presently corresponds to the state of the art and is in excellent agreement with the best experimental storage ring measurements available at this time [3–5].

The theory presented in Kokoouline *et al*. [1] and dos Santos *et al.* [2] is based on *ab initio* computations of bound *n*=4 Rydberg states of H_{3} by Mistrík *et al.* [6], from which the static linear JT splitting was extracted, using a procedure initiated by Staib & Domcke [7] (who based their work on earlier *ab initio* computations of Nager & Jungen [8] for *n*=3). This information then subsequently served to compute the dynamical electron capture process by the positive ion, followed by neutral molecular dissociation [2]. The fact that it is possible to predict continuum (collision) processes on the basis of structural properties of bound states is remarkable. It derives from the recognition that electronically excited bound molecular states are in fact ‘precursors’ [9] of the electron–ion scattering states, with the result that their quantum defects imply the values of the continuum electron phase shifts. This fact has been highlighted in the recent work of Jungen & Pratt [10] on the DR of H, where spectroscopic information on the H_{3} 3*pe*′ Rydberg states from Herzberg and co-workers [11–13] was combined with an analytical expression for the DR cross section to provide a simple but quantitatively rather accurate description of the process.

The expression derived in Jungen & Pratt [10] gives the DR cross section in instances where the resonant electron capture process by vibronic interaction is the rate determining step of the DR process. The DR cross section averaged over the capture resonances then reads
1.1
All quantities in equation (1.1) are in standard units so that the cross section will be in m^{2}. *k* is the wavenumber of the incident electron in m^{−1} and *n** is the effective principal quantum number of the Rydberg state *n* from which the JT interaction is extracted. The expressions in the two brackets (…) are dimensionless and actually express two characteristic energies in atomic units: *ω*_{2} is the vibrational frequency of the active non-totally symmetric JT mode, while *D*_{n}*ω*_{2} is the depth of the potential trough produced by static JT distortion in the potential surface of the *npe*′ Rydberg state, which serves for the determination of the JT interaction. *D*_{n} is the dimensionless linear JT parameter, while is the Rydberg constant. Note that quantum defect theory predicts *D*_{n}*ω*_{2} to scale approximately with *n**^{−6} [10]. This compensates the factor *n**^{6} in equation (1.1), in such a manner that 〈*σ*〉 should basically be independent of the spectroscopic state *n* from which *D*_{n} is extracted. In this paper, we consider primarily *n*=3 with *n**=2.594. Recall also that *k*^{2} is proportional to the incident electron energy, *E*. One has
1.2
which means that when *k*^{2} is taken as the energy in Rydberg units, the cross section will be obtained in units bohr^{2}.

The interplay between excited state structural properties and molecular spectroscopy, on the one hand, and the dynamical reactive collision process, on the other hand, is thus a recurrent theme in the theory of DR of H. Indeed, the theoretical approaches employed in both dos Santos *et al*. [2] and Jungen & Pratt [10] are derived from multi-channel quantum defect theory [14–17], which systematizes the basic unity existing between excited state spectroscopy and low-energy collision physics.

In this contribution, we review the various determinations of the JT parameters in the 3*pe*′ pair of H_{3} Rydberg states that have been made. We show to what extent they are compatible with one another—the answer to this question is not obvious from cursory inspection of the various papers because different coordinate systems are used in different papers to represent the vibrational motion, and the relationship between the various quantities discussed by different authors is not always quite clear. We also attempt to redetermine the linear and quadratic JT splitting constants on the basis of the most sophisticated *ab initio* H_{3} 3*pe*′ potential surfaces existing to date. Although King & Morokuma [18] extracted a quadratic JT constant from their *ab initio* surface, to date, no quadratic constant for the later *ab initio* surfaces [6,8,19] has been reported; similarly, no quadratic constant has been extracted from the spectroscopic data of Herzberg *et al.* [11,12]. While we focus in this work on the 3*pe*′ state of H_{3}, we also consider the *n*=4 and *n*=5 members of the *npe*′ Rydberg series, for which *ab initio* potential energy surfaces also exist [6]; consideration of these surfaces provides information on the energy dependences of the quantum defect parameters.

## 2. Theoretical considerations

The initial first-principles calculation of H_{3} Rydberg states appears to be that by King & Morokuma [18], who used Koopman’s theorem to compute the surfaces of 15 Rydberg states of H_{3} for geometries near the H minimum. This work was soon followed by the computations of Nager & Jungen [8], who used their frozen-core Rydberg *ab initio* method to predict numerous Rydberg states over a wide range of geometries. More recent potential surfaces of H_{3} Rydberg states have been published in Mistrík *et al*. [6] and Galster *et al*. [19]. Among these, the highly correlated electronic wave functions and energies [6] obtained with the multi-reference code of Fink & Staemmler [20] based on the coupled electron-pair approximation (CEPA) probably remain the most accurate values available today, and we shall use them below in an attempt to reevaluate the JT parameters of the *npe*′, *n*=3−5 Rydberg states.

### (a) Bond lengths versus symmetry-adapted coordinates

The *ab initio* work on excited states of H_{3} and H has been reported alternatively in terms of Dykstra coordinates [6,8,21], hyperspherical coordinates [19] or mass-scaled Jacobi coordinates [22]. In all these papers, the relationships between coordinates employed and the straightforward bond-length coordinates *R*_{1},*R*_{2},*R*_{3} have been spelled out in detail. For the purposes of the present discussion, we therefore start out from the bond lengths and their changes. Following the classic book by Herzberg [23], we refer to the changes of the lengths of the three sides of the triangle as *Q*_{1}, *Q*_{2} and *Q*_{3} (where *Q*_{1} refers to the change of the distance between atoms 2 and 3, etc.). In the following, *Q*_{1}=*Q*_{2}=*Q*_{3}=0 is taken to correspond to the H equilibrium configuration with *R*_{1}=*R*_{2}=*R*_{3}=1.6504*a*_{0}. Symmetry-adapted coordinates *S*_{i} (*i*=1−3) appropriate for the *D*_{3h} triangular symmetry are then introduced using familiar procedures that yield the relationships
2.1
For small amplitude *Q*_{i} around the equilateral triangular geometry and for a non-degenerate electronic state, the vibrational motion then becomes separable in terms of the three symmetry modes with an identical effective mass 3*M* for each of them (where in our case *M* is the proton mass). It is customary to use polar coordinates for the degenerate non-totally symmetric mode, by setting
2.2
Here, *ϕ* is the coordinate that describes the so-called pseudo-rotational motion.

### (b) Jahn–Teller surfaces for weak distortion

For a doubly degenerate electronic state, in terms of these coordinates and based on arguments developed by Longuet-Higgins [24], the 3×3 potential matrix is established to have the form
2.3
(A condensed account of the considerations leading to equation (2.3) can be found in Jungen & Pratt [10].) Diagonalization of the matrix equation (2.3) yields the real electronic Born–Oppenheimer potential surfaces *U*_{1}(*S*_{1},*ρ*,*ϕ*) and *U*_{2}(*S*_{1},*ρ*,*ϕ*),
2.4
The potential surfaces equation (2.4) exhibits the familiar features of JT distortion [24]: (i) for *ρ*=0 (equilateral triangle) a conical intersection occurs, (ii) the linear JT term lowers the potential minima by an amount *f*^{2}/2*k*_{ρρ}, (iii) the potential minimum corresponds to a distorted configuration Δ*ρ*=*f*/*k*_{ρρ}, and (iv) the quadratic term introduces a *ϕ* dependence in the potential that yields three minima in the lower potential trough. Figure 1 illustrates how the dimensionless JT parameter, *D*, measures the lowering of the minimum by the linear term in units of the vibrational frequency *ω*_{2}, i.e. *Dω*_{2}=*f*^{2}/2*k*_{ρρ}, and how the quadratic term, *g*, produces three secondary minima lying *g*(Δ*ρ*)^{2}/2 below the minima of the linear form and twice as much below the saddle points separating the minima. (Note incidentally that *Dω*_{2} is an electronic quantity independent of nuclear mass, which means that the isotope dependence of the vibrational frequency is exactly compensated by that of *D*. As a result, the isotope dependence of equation (1.1) is determined solely by the isotope dependence of *ω*_{2}.) The additional lowering of the potential minima owing to the quadratic term is sometimes [24] expressed as *g*(Δ*ρ*)^{2}/2=4*qDω*_{2}, where the dimensionless quantity *q*=*g*/4*k*_{ρρ} is the so-called quadratic JT parameter.

For small amplitudes *ρ*, the real Born–Oppenheimer angular factors corresponding to the potential surfaces of equation (2.4) and resulting from the diagonalization of equation (2.3) are given by
2.5
Here, e^{±iΛϕ} with *Λ*=1 are the angular factors of the electronic wave functions representing the degenerate p orbital components for *ρ*=0 that lie in the molecular plane and form a *pe*′ Rydberg state. Note how the functions of equation (2.5) resulting from the diagonalizing equation (2.3) exhibit the effect of Berry’s phase [25]: they correspond to *Λ*=1/2 instead of *Λ*=1 and change sign under the identity symmetry operation .

### (c) Analysis of *n*=*3* ab initio potential surfaces

We have used the *ab initio* 3*pe*′ surfaces from Mistrík *et al*. [6] and Galster *et al*. [19] in order to reexamine the JT parameters relevant for the *npe*′ Rydberg channel. This task has been performed previously by Staib & Domcke [7] on the basis of the theoretical surfaces from Nager & Jungen [8], and more recently by Mistrík *et al*. [6] based on the new potential surfaces computed there. As mentioned above, the most complete theoretical evaluation of DR of H, owing to dos Santos *et al.* [2], is based on the JT parameters for *n*=4 from Mistrík *et al*. [6].

We are interested at this stage in the shapes of the 3*pe*′ potential surfaces near the conical intersection point; therefore, we limit our analysis to geometries with energies that do not lie more than 0.02 a.u. above the energy corresponding to the intersection. We take the energy of the latter as the origin, i.e. *W*_{0}=0 in equation (2.4). Table 1 lists the 22 out of 28 geometries computed in Mistrík *et al.* [6] for which this criterion is satisfied. The H potential is known to be highly anharmonic [26]. Following King & Morokuma [18] and in order to get a reasonably realistic representation of the potential energy surfaces, we therefore included cubic potential terms and on the diagonal of equation (2.3). These cubic terms were determined from the energies computed in Mistrík *et al.* [6] for H and were kept fixed in the subsequent H_{3}(3*pe*′) calculations (table 2). The force constants *k*_{11} and *k*_{ρρ} were also determined initially for the ion, and we subsequently determined the change Δ*k* in the 3*pe*′ Rydberg state, i.e. *k*=*k*^{+}+Δ*k* (*k*^{+} ion value).

During the calculations, it soon became clear that the 3*pe*′*ab initio* potential points exhibit some scatter, possibly due to slight differences of the convergence behaviour of the CEPA calculations for different geometries. We therefore employed the ‘robust’ weighted least-squares procedure of Ruckstuhl *et al*. [27] as implemented by Dabrowski *et al*. [28]. Outlying data points are weighted down in a manner that is adjusted dynamically as the least-squares iterations are progressing. Table 2 presents the results of the least-squares fits. The residuals of the fit are given in the last column of table 1. The overall standard deviation of the fit amounts to 5.8×10^{−5} a.u. (or 13 cm^{−1}), whereas the mean deviation for unit weight is 0.81×10^{−3} a.u., i.e. nearly 15 times more. Inspection of the last column of table 1 indicates that two outliers among the 44 data points are the principal cause for this large difference. Note that, of course, the standard deviations given here refer to the fitting procedure employed and not to the absolute accuracy of the *ab initio* calculations.

We have attempted to determine the JT splitting parameters also from the more recent *ab initio* 3*pe*′ potential surfaces of Galster *et al*. [19]. The linear term *f* thus obtained is compatible with our value given in table 2, but its uncertainty turns out to be more than an order of magnitude larger. In addition, we have not been able to determine the quadratic term *g* from the data of Galster *et al*. [19]. The reasons for this appear to be the facts that fewer geometries were considered near the conical intersection in Galster *et al*. [19], and that none of the computed energies lies below that corresponding to the conical intersection. By contrast, table 1 shows that the four geometries included in Mistrík *et al.* [6] yield potential energies lower than the intersection point, and thus help to characterize the JT induced potential trough.

### (d) Analysis of higher potential energy surfaces

In addition to potential energy surfaces corresponding to *n*=3, Mistrík *et al.* [6] also provide analogous data relating to the 4*pe*′ and 5*pe*′ JT pairs of states. While the quantum-chemical computations of these states are expected to be somewhat less well converged, they do provide interesting information about the energy dependence of the relevant JT parameters. This fact has already been pointed out and exploited in Mistrík *et al.* [6].

We reanalyse these data here in a one-channel quantum defect approach inspired by equation (2.4), applied separately to each JT component and molecular geometry. Thus, for instance, the origin *W*_{0} in equation (2.4) is replaced by the quantum defect, *μ*_{np}, corresponding to the H ion equilibrium geometry, and the linear and quadratic parameters *f* and *g* are replaced by their quantum defect counterparts *μ*_{f} and *μ*_{g}, respectively. The input data consist now of the Rydberg energies *ϵ*_{n}=−(*n**)^{−2} (in Rydbergs) derived from Mistrík *et al.* [6] for *n*=3, 4 and 5, and there is no need for fitting the ion potential surface. Otherwise, the fitting procedure is exactly analogous to that described in §2*c*. It has not been possible to extract the quadratic JT quantum defect *μ*_{g} from the data for *n*=4 and 5.

Figure 2 is an Edlén plot [29] showing the quantum defect parameters obtained for *n*=3−5 plotted versus the electron energy *ϵ*. For *n*=3, values directly comparable to those given for *f* and *g* in tables 1 and 2 may be derived from the corresponding quantum defects *μ*_{f} and *μ*_{g} simply by division by (*n*−*μ*_{np})^{3}. Note that the parameters *μ*_{k11} and *μ*_{kρρ} account only for the bond effect owing to the Rydberg electron—in analogy with the parameters Δ*k*_{11} and Δ*k*_{ρρ} determined earlier—and do not include the force constants of the ion. Figure 2 demonstrates that all the quantum defects exhibit a linear dependence with energy as generally expected in Rydberg systems. A linear regression yields the values *μ*(*ϵ*=0) and the slopes ∂*μ*/∂*ϵ*, cf. the inset of the figure.

## 3. Discussion

### (a) Jahn–Teller parameters

It is instructive to compare the JT terms for *n*=3 from table 2 with those derived from high-resolution emission spectroscopy in the 1980s [11,12], as well as with those derived previously by various authors from quantum-chemical potential surfaces [6,7,8,18]. This requires converting the definitions and coordinates used in previous papers to the ones used here. The comparison is made in table 3. As may be seen from the table, a reassuring outcome of the present analysis is the fact that the linear JT (*f*) splitting parameter values determined by various authors are all consistent to within about 10 per cent. Moreover, the *f* value extracted by Staib & Domcke [7] from the frozen-core surface of Nager & Jungen [8] is found to agree to within 1 per cent with the value derived here from the correlated calculation of Mistrík *et al.* [6]. Judging from equation (1.1), we conclude that the corresponding DR cross sections should agree to within 2 per cent. The linear JT term therefore is apparently not governed by electron correlation effects. Note incidentally that the same is not true for the quantum defect at the conical intersection: while the frozen-core calculations of Nager & Jungen [8] predicted a value *μ*=0.346, the determination of Mistrík *et al.* [6] derived from a highly correlated electronic wave function yielded *μ*=0.406, that is, a significantly increased value corresponding to a lowering of the 3*pe*′ pair of states by 700 cm^{−1}.

Similarly, the values for the depth of the JT-induced potential minimum determined by Herzberg *et al.* [12], and more recently by Vervloet & Watson [13], also turn out to be consistent with the various quantum-chemical determinations. Herzberg *et al*. [12] and Vervloet & Watson [13] derived their values from the observed rotational structure of the 3*pe*′ manifold of levels combined with the vibrational frequency of the 3*pe*′ Rydberg state predicted in the study of King & Morokuma [18]. The value 86.1±0.7 cm^{−1}, which we obtain here for the same state, is in quite good agreement with the values given there, *viz*. 87.0 cm^{−1} [12] and 89.1 cm^{−1} [13]. The value cm^{−1} used in our previous paper [10] for evaluating equation (1.1) is seen to be about 15 per cent too small (cf. table 3).

The determination of the quadratic JT parameter in the 3*pe*′ state is new. It allows us to predict the splitting of the *A*_{1}′ and *A*_{2}′ vibronic levels, which arises for in the *v*_{2}=1 JT overtone mode, and should become observable if new spectroscopy experiments are carried out. Here, is the quantum number associated with the sum of the nuclear (*l*) and electronic () orbital angular momenta. The splitting amounts to *ω*_{2}*g*/*k*_{ρρ}≡4*qω*_{2} [24] (where *q* (=0.00256 here) is the dimensionless quadratic JT parameter, see §2*b*), so that with the parameters from table 2, we predict a value of 25.8 cm^{−1}.

### (b) Implications for dissociative recombination

We have confirmed here that all the previous analyses of *ab initio* surfaces [6–[8],18,19] yield mainly consistent predictions of the DR cross section of H. The question nevertheless arises which values should best be used for the theoretical predictions. The DR process involves a slow continuum electron so that, in principle, for instance in equation (1.1), the value *n**^{6}*D*_{n}*ω*_{2} extrapolated to threshold () should be used. The value *μ*_{f}(*ϵ*=0)=0.399 determined here (cf. figure 2) would imply a value *f*=0.0229 for *n*=3, i.e. 14 per cent less than given in table 2 (0.0265), so that the DR cross-sectional prediction decreases by 25 per cent if the extrapolation to threshold is made based on the data of Mistrík *et al.* [6]. This error may be reduced to about 15 per cent if one considers that the capture process couples the threshold energy electron primarily to Rydberg states with *n*≈8. However, one may also argue that the true energy dependence of the JT parameter *μ*_{f} might be masked partially because of a less complete convergence of the *ab initio* calculations for higher *n* states when compared with *n*=3. More theoretical and/or experimental work may be required to answer this question definitively, but at this stage, it appears that the energy dependences of the relevant quantum defect parameters are relatively mild, and that the data for *n*=3 may be the safest to use as only for *n*=3 has a detailed comparison between various independent experimental and theoretical data so far been possible.

In our previous work [10], only the off-diagonal matrix elements linear in *ρ* were considered. In a harmonic oscillator basis, these matrix elements are only non-zero for Δ*v*=±1, so that for collision energies greater than one vibrational quantum, capture is not possible and the recombination cross section goes to zero if the assumptions underlying equation (1.1) are correct. The extraction of the quadratic JT parameter allows the consideration of processes with greater values of Δ*v*. In particular, the quadratic *ρ*^{2} terms considered here couple vibrational levels with Δ*v*=±2, allowing capture at energies up to two vibrational quanta. Evaluation of the corresponding matrix elements [30] indicates that the capture cross section for the quadratic Δ*v*=2 process is only approximately 0.1 per cent of that for the Δ*v*=1 process.

From these considerations, it follows that above the threshold, the DR cross section should fall by a factor of approximately 1000 if only the capture process deriving from the quadratic JT effect is active in this region. However, the experimental cross section appears to fall by only about one order of magnitude at the threshold (cf. [5], fig. 7). In the absence of significant interaction, alternative mechanisms above the threshold are (i) electron capture via continuum–continuum interaction between the open and channels, followed by dissociation, or (ii) electron capture into *v*^{+}=2 Rydberg states via two Δ*v*^{+}=1 steps, followed by dissociation. These mechanisms however only become effective if, unlike assumed by equation (1.1), dissociation is not so much faster than autoionization.

If dissociation is not so much faster than autoionization, there will be a partial breakdown of equation (1.1) at low energy, below the threshold, as reionization may occur after capture and thus reduce the cross section from the value of equation (1.1). However, we know [31] that even if the autoionization and predissociation rates of the quasi-bound Rydberg levels become equal so that significant reionization occurs, the averaged DR cross section is reduced by a factor of 2 at most, which will not be very visible on logarithmic plots, such as those used in McCall *et al*. [3] and Kreckel *et al*. [4,5].

Finally, we wish to point out that while the experimental cross sections of McCall *et al*. [3] and Kreckel *et al*. [4] show a significant drop by a factor of approximately 10 at the threshold (0.313 eV), careful inspection of the experimental curves indicates that they also show a second, albeit smaller, drop at approximately the energy of the threshold (0.620 eV) of an additional factor of approximately 2. This suggests that capture by Δ*v*_{2}=2 does make a non-negligible contribution to the cross section.

## 4. Conclusion

The linear and quadratic JT parameters have been extracted from the theoretical potential surfaces of Mistrík *et al.* [6] for the 3*pe*′ Rydberg states of H_{3}. The linear JT parameter, *f*, is in very good agreement with values determined previously from the same surfaces, from a second set of theoretical surfaces and from experiment. The quadratic JT parameter, *g*, determined here had not been extracted from the present surfaces previously, and is a factor of approximately 3 smaller than the value determined from the surfaces of King & Morokuma [18]. The present results are reassuring in that they demonstrate both the quality of the potential energy surfaces and the consistency of previous experimental and theoretical determinations of the JT parameters for the 3*pe*′ state of H_{3}. This could indeed be inferred already from our previous work [10], where it was shown that the analytic expression equation (1.1) produces cross sections that agree very well with the all degrees-of-freedom computations of dos Santos *et al*. [2]. The present analysis confirms this and eliminates the possibility that this agreement could have been fortuitous.

Figure 2 indicates that the dependence on energy of the linear JT quantum defect parameter *μ*_{f} is not very strong, although the quality of the quantum-chemical calculations for *n*>3 remains to be verified. It might account for a lowering of up to 15 per cent of the DR cross section, but which, concerning our own previous work [10], would be compensated by the fact that the value *D*_{n}*ω*_{2} used there was about 15 per cent too small (see §3*a*).

The determination of the quadratic JT parameter allows the assessment of capture processes with Δ*v*>1. However, Δ*v*=2 processes driven by the quadratic JT interaction have been found here to have a minimal effect on the DR cross section. It is also interesting that although the experimental DR cross section drops by an order of magnitude at the threshold, the expected theoretical drop based on vibrational capture into neutral Rydberg states is significantly larger than that.

As a final more general remark, we may say that the *height* of drops of DR cross sections near vibrational thresholds is an interesting feature (in addition to their very *appearance*, which is a general phenomenon [31]), as it may contain information on the relative rates of dissociation versus ionization processes. This is the subject of a forthcoming publication of Schneider *et al*. [32].

## Acknowledgements

Ch.J. thanks the ANR (France) for financial support under contract 09-BLAN-020901. He also received partial support from the Miescher Foundation (Basel, Switzerland). S.T.P. was supported at Argonne by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under contract no. DE-AC02-06CH11357. We finally wish to thank the referees for their constructive comments.

## Footnotes

One contribution of 21 to a Theo Murphy Meeting Issue ‘Chemistry, astronomy and physics of H

_{3}^{+}’.

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