## Abstract

The process of resonant ion-pair formation following electron collisions with is studied. The relevant diabatic potential energy surfaces and the electronic couplings between these surfaces are calculated. The reaction is then described using a time-dependent approach with wave packets propagating on the coupled potentials. In order to describe the reaction, it is found necessary to include at least two dimensions in the model. The effects of the Rydberg states on the cross-section for this process are discussed.

## 1. Introduction

During the last decades, dissociative recombination (DR) of at low electron collision energies has been the object of great interest and much study by both experimentalists and theoreticians. The major focus has been to explain the relative large DR rate observed for despite the lack of a direct mechanism. Calculations by Kokoouline *et al*. (2001) showed that the complete explanation required a study in full-dimensionality and the inclusion of the Jahn–Teller effect in H_{3} to correctly describe the capture mechanism of the recombining electron and subsequent dissociation dynamics.

At higher collision energies, it has long been established that the electron can be captured directly into doubly excited neutral states. The lowest A_{1} resonant state (*C*_{2v} symmetry) was first calculated in one dimension by Kulander & Guest (1979) and later by Michels & Hobbs (1984). These calculations showed that this state crosses the ionic groundstate near the third vibrational level, and hence cannot explain the low-energy DR. Using a diabatic representation, the lowest resonant state will dissociate into the ion-pair fragments, , after first crossing the manifold of Rydberg states. Since the ion-pair state will couple to the Rydberg states by electronic couplings, it was proposed that both neutral and ionic fragments would be formed during the dissociation. Using the complex Kohn variational method, Orel *et al*. (1994) calculated the positions and autoionization widths of the four lowest resonant states of H_{3}.

The DR of at higher energies was first studied experimentally using the CRYRING ion-storage ring, where the neutral fragments were detected (Larsson *et al*. 1993). The measurement showed a peak in the cross-section, where the energy is high enough to allow capture into the resonant states predicted by the theoretical studies. In the same year, Orel & Kulander (1993) used the calculated resonant states to explain the observed high-energy peak in the DR cross-section. In this two-dimensional study, it was assumed that everything that is captured into the resonant states will form neutral fragments. The dissociation dynamics after the resonant states have crossed the ion potential and the couplings between the resonant states were neglected.

The cross-section of ion-pair formation has been measured using an inclined beam measurement (Peart *et al*. 1979), single-pass merged beams (Yousif *et al*. 1993) and the CRYRING ion-storage ring (Kalhori *et al*. 2004). In all the measurements, only the H^{−} fragments were detected and it is not possible to separate the channel with threshold *E*_{th}=5.4 eV from H+H^{+}+H^{−} with *E*_{th}=8.1 eV. The three measurements show cross-sections with different energy dependencies possibly due to difficulties in the former two measurements in controlling the vibrational distribution of the ions. However, when the vibrational distribution is suppressed by collisional quenching, all the three measurements show a cross-section with a magnitude of *ca* 2×10^{−18} cm^{2}. There is no sign of any interference structure in the measured cross-sections.

The goal of the present study is to obtain a better understanding of the ion-pair formation in collision of electrons with . To describe the reaction, it is necessary to have a complete description of dynamics, including all couplings between the states. We will first describe how the relevant diabatic potential energy surfaces and the electronic couplings among them are calculated. This is followed by a description of the wave packet dynamics and then the results from the one- and two-dimensional studies are discussed.

## 2. Potential energy surfaces and couplings

The adiabatic resonant states and autoionization widths were calculated by Orel *et al*. (1994) using the complex Kohn variational method. In order to transform from the adiabatic resonant states to the corresponding diabatic states, the geometry dependence of the configuration interaction (CI) coefficients of the dominant configurations is needed. This is obtained by performing restricted CI calculations, where the lowest molecular orbital (1*a*_{1}) is kept singly occupied in order to avoid contributions from the ionization continuum. Using the CI coefficients, the adiabatic *A*_{1} potentials are transformed to the corresponding diabatic potentials and the electronic coupling between them is obtained by assuming that the potentials couple two-by-two. The ion-pair state is defined from the (1*a*_{1})^{1}(2*a*_{1})^{2} configuration (Larson & Orel 2001). The adiabatic states of H_{3} situated below the ion are calculated using a full CI calculation, where each H-atom is represented by a (8s, 5p, 1d) basis and the wavefunction contains 63 000 configurations. Again the CI coefficients are used to transform to the diabatic potentials and couplings. For the curve crossings at small internuclear distances with higher Rydberg states (*n*≥3), the coupling between the states is estimated using a scaling of the electronic coupling between the resonant state and the ionization continuum (Giusti-Suzor *et al*. 1983). Note that only the couplings to the ion-pair state are included. The couplings between the second resonant state and the Rydberg states or the couplings among the Rydberg states are not included in the model. The calculated diabatic potentials are shown in figure 1, where the H–H distance is kept at *r*=1.65*a*_{0}. The distance from the centre of mass of the H–H pair and to the third H-atom is denoted by *z*.

## 3. Wave packet dynamics

When the potentials and couplings are calculated, the dissociation dynamics is studied using wave packets propagating on the coupled potentials. First, the wave packets are initiated on the two resonant states *i*=1, 2,(3.1)where *Χ*_{v=0} is the vibrational wave function of the ion and *Γ*_{i} is the autoionization width of the resonant states (McCurdy & Turner 1983). Here, the initial capture into the vibrational excited Rydberg states is neglected. This approximation is plausible at the high-energy region relevant for this reaction. The wave packets are then propagated by a direct integration of the time-dependent Schrödinger equation using the Chebyshev propagator (Tal-Ezer & Kosloff 1984). The diagonal elements of the Hamiltonian contain the kinetic energy and the diabatic potentials and the off-diagonal elements contain the electronic couplings. Autoionization from the resonant states is included by letting the potentials be complex *V*_{i}=*E*_{i}−*iΓ*_{i}/2 above the ion-potential. In total, we include the two resonant states and then four Rydberg states, where the highest Rydberg state has an effective coupling to the ion-pair state to account for the coupling to the infinite number of Rydberg states not included in the model. The wave packets are propagated out in the asymptotic region, where all couplings are zero. The cross-section for ion-pair formation is then obtained by analysing the dissociating flux through a plane with *z*=*z*_{stop}. In the present calculation, we used *z*_{stop}=15.0*a*_{0}, where we assume the asymptotic region is reached. The dissociating flux is calculated from the half-Fourier transform of the overlap between the wave packet on the ion-pair state and the vibrational eigenfunctions (*ϕ*_{v}(*r*)) of the *E*_{1}(*r*, *z*_{stop}) potential ( eigenfunctions),(3.2)and the cross-section is given by(3.3)Here, *μ*_{z} is the reduced mass for the *z*-coordinate and *κ*_{v} is the wavenumber of the dissociating fragments for a given vibrationals level *v* of (for details, see Balint-Kurti *et al*. 1990, 1991; Haxton *et al*. 2004).

## 4. Results and discussion

To simplify the treatment, we start by performing a one-dimensional model of the reaction, where *r*=1.65*a*_{0} is kept frozen at the equilibrium distance of the ion. Details on the one-dimensional study have been published in Kalhori *et al*. (2004); we summarize the results in figure 2. The solid curve in figure 2 shows results when we include only the diabatic ion-pair state (in all calculations it is assumed that the ion initially is in the *v*=0 vibrational level). The measured cross-sections of ion-pair formation from the CRYRING experiment (Kalhori *et al*. 2004) are shown with filled squares. The calculated cross-section is peaked at lower energies with a much sharper threshold and is *ca* 7.5 times larger than the experimental cross-section. The dashed curve in figure 2 shows the results when the resonant state and the couplings between the two resonant states are included. The double-peak structure (not observed in the measured cross-section) can be explained as an interference effect between the two resonant states. The results, including the couplings to the Rydberg states at small internuclear distances, are shown by the dotted curve and the results then adding the couplings at large distances are shown by the dashed-dotted curve. The inclusion of coupling to the Rydberg states causes the magnitude of the cross-section to decrease, but it is still approximately a factor 5 larger than the experimental cross-section and peaked at lower energies.

The next step is to perform a two-dimensional study of this reaction. From the electron scattering calculations by Orel *et al*. (1994), it was found that the resonant states are strongly repulsive as functions of the two Jacobi distances, but almost flat when the angle is varied. Therefore, we assume that the angle can be kept constant at 90° during the dissociation. From the *ab initio* calculations, the diabatic potentials and couplings in the range 1.0*a*_{0}≤*r*≤3.5*a*_{0} and 1.0*a*_{0}≤*z*≤15.0*a*_{0} are calculated and these are then extrapolated out to have the correct asymptotic behaviour. The first step is again by propagating the wave packet on the diabatic ion-pair state alone. The resulting cross-section is shown by the solid curve in figure 3 and it shows a peak centred at lower energies than the experimental cross-section, but with a better threshold behaviour compared with the one-dimensional study. The magnitude is still approximately a factor 5 larger than the experimental cross-section.

In the next step, the second resonant state is included. The inclusion of the second dimension washes out the interference effects that gave the double-peak structure in the one-dimensional study. The resulting cross-section is very similar to the cross-section obtained with the ion-pair state alone. Finally, the effects from the Rydberg states on the ion-pair cross-section are examined. In the one-dimensional study, the inclusion of the Rydberg states dropped the cross-section with *ca* 40%. If we assume the same is true for two dimensions, we get the cross-section shown with the dotted curve in figure 3. The cross-section is still larger than the measured cross-section and peaked at smaller energies. An alternative method to estimate the loss to the Rydberg states is by using the Landau–Zener models and calculate the survival probability along the diabatic ion-pair state. Since the Landau–Zener model is a one-dimensional model, we use the classical trajectory in the diabatic ion-pair state as a reaction coordinate. The cross-section is now smaller than the measured cross-section. However, the flux that is transferred to a Rydberg state at the inner curve crossing may jump back to the ion-pair state at the curve crossing at larger distances. When this effect is included, the ion-pair cross-section increases slightly. It is still smaller than the experimental cross-section and peaked at lower energies. However, it should be remembered here that we include only one of the channels, , measured in the experiment. To summarize, we can conclude that in order to describe the dynamics of this reaction, we need to include at least two dimensions, and the second dimension will smear out the interference effects. The role of the Rydberg states is to reduce the flux dissociating into the ion-pair fragments. However, to obtain a complete picture of the dynamics, all the states, the resonant states and Rydberg states, have to be included in the wave packet propagation. We can then also estimate the vibrational distribution of the fragments and we can study the reaction for different isotopologues.

## Acknowledgments

Å.L. acknowledges support from The Swedish Research Council and the Göran Gustafsson Foundation and A.E.O. acknowledges support from the National Science Foundation, grant no. PHY-02-44911.

## Footnotes

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

_{3}^{+}’.

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