From quantum chemistry to dissociation kinetics: what we need to know†

The relationship between rate constants for dissociation and the reverse association reactions and their potential energy surfaces is illustrated. The reaction systems e− + SF6 ↔ SF6− →SF5− + F, H + CH3 ↔CH4, 2 CF2 ↔ C2F4, H + O2 →HO2, HO + O ↔HO2 ↔ H + O2, and C + HO →CHO are chosen as representative examples. The necessity to know precise thermochemical data is emphasised. The interplay between attractive and anisotropic components of the potentials influences the rate constants. Spin–orbit and electronic–rotational coupling in reactions between electronic open-shell radicals so far generally has been neglected, but is shown to have a marked influence on low temperature rate constants.


Introduction
Dissociation and the reverse association reactions play an important role in combustion, atmospheric and interstellar chemistry, as well as in many other gas-phase reaction systems. While the energy levels of the reacting species generally are well known from high-resolution molecular spectroscopy and quantum chemical calculations, the dissociation and association rates are much less precisely measured and rate theories are generally far from being satisfactory. By presenting a few examples, this essay describes the state of the art and highlights areas where priority needs for determining rate parameters can be identified.
Electron attachment to SF 6 and the reverse detachment from SF 6 − in competition to the dissociation of SF 6 − is chosen as the first example. This system is of considerable fundamental and practical importance and it has been studied in detail over some time. Many characteristic features of plasma chemical kinetics are encountered here and progress in the understanding can be documented particularly well. Thermal dissociation of methane is of similar importance in hydrocarbon pyrolysis and oxidation and hence in combustion chemistry. This system has been chosen because theoretical studies of its elementary processes -collisional energy transfer between the excited molecule and its surrounding bath gas as well as intramolecular bond breaking and bond formation -are reaching a mature state without the necessity for empirical adjustment of parameters. The thermal dissociation of perfluoroethene and the reverse dimerisation of difluorocarbene are described for comparison, again illustrating the present understanding, but in addition emphasising the basic differences between hydro-carbon and fluorocarbon dissociation and association kinetics. Finally, the 'simple' reaction system H + O 2 ↔ HO 2 ↔ HO + O is analysed with respect to various aspects of dissociation and association dynamics. The open-electronic shell character of the reactants renders a quantitative analysis of the dynamics particularly difficult. The effects to be expected are illustrated by a comparison with the C + HO association process for which the consequences of spinorbit coupling have already been analysed. The interplay between attractive and anisotropic components of the potential energy surface, as in the CH 4 and C 2 F 4 systems, is illustrated also for the HO 2 system with respect to its influence on the dissociation and association dynamics.
In summary, this report describes which properties of potential energy surfaces and which details of the reaction dynamics on these potentials need to be known with priority in order to arrive at least at a semiquantitative understanding of dissociation and the reverse association processes.

2.
Kinetics of the e − + SF 6 ↔ SF 6 − → SF 5 − + F reaction system The inert, non-toxic, and insulating molecule SF 6 is used in many technical applications. For example in the plasma etching of silicon chips the reaction SF 6 + e − → SF 5 − + F acts as a source for fluorine atoms which may remove unprotected silicon in a reaction 4 F + Si → SiF 4. By acting as an 'electron sponge', SF 6 efficiently reduces the concentration of mobile electrons from ionised gases. Because of this ability, SF 6 may be used to shape switching arcs in circuit breakers of power plants and here finds wide practical applications. While it is a very useful substance, one has to ask whether it is harmless. It is a powerful greenhouse gas, being 20,000 times more effective (per weight) than CO 2 with an atmospheric increase of 4% per year. Its atmospheric lifetime is more than 1000 years; its removal may involve photolysis or the considered reaction e − + SF 6 → SF 5 − + F. For the described reasons, the large interest in the kinetics of this reaction system thus is understandable.
The first question to ask obviously is about the thermochemistry of the reaction, in particular about the electron affinity E a of SF 6 , i.e. the energetics (at 0 K) of the reaction e − + SF 6 → SF 6 − . Surprisingly, this quantity for a long while could be estimated only by indirect methods [1] giving E a = 1.05 eV, a photoelectron spectrum including the 0-0 transition not being easily accessible. However, the derived value of E a differed considerably from quantum chemical results. For example CCSD(T)/6-311 + G(2df) calculations led to [2] E a = 0.92 eV. In this situation, a more direct experimental access to E a was searched and finally found in the measurements of the thermal rate constants for electron attachment e − + SF 6 → SF 6 − and the reverse electron detachment [3]. Although only a small temperature range could be covered for detachment (590 to 670 K), the equilibrium constant K c in this range became accessible. A third law analysis of K c could be made, using vibrational frequencies from MP2/aug-cc-pvdz calculations. The result of this analysis was a value of E a = 1.20 eV, the discrepancies thus remaining unsolved. Over the last years, the problem finally found a surprising solution. First, CCSD(T) calculations with larger AO basis sets [4] than used before on the one hand confirmed the small value E a = 0.94 eV; on the other hand, a distorted C 4ν geometry of SF 6 − was found in contrast to earlier conclusions about an O h geometry. In addition, highly anharmonic unseparable vibrations were obtained. This prompted [5] a revised third law analysis of the experimental K c , now leading to E a = 1.03 ± 0.05 eV, which was in agreement with the earlier less direct determinations, but still far off the much lower quantum-chemical value. The solution of the dilemma finally was reached by new quantumchemical calculations including a series of higher order (such as scalar-relativistic, inner-shell, and post-CCSD(T), and other) corrections [6]. It was found that, instead of cancelling, these added up, moving the calculated value from E a = 0.94 to 1.0340(±0.03) eV. The described studies, finally leading to quite satisfactory agreement between experiment and theory, illustrate the large effort sometimes required to obtain at least the thermochemical basis for a proper analysis of dissociation/association kinetics. (It should be noted that most recent studies again questioned the agreement [7]. By analysing lifetime distributions of particularly long-lived anions in cooled ion traps, a value of 0.91 eV close to the uncorrected quantum-chemical value of 0.94 eV was derived such that further work is needed again.) Figure 1. Falloff curves of the rate constants k at for nondissociative electron attachment to SF 6 (modelling from Ref. [8]; dashed lines: only collisional stabilisation, full lines: collisional plus radiative stabilisation of SF 6 − * ).
The attachment of electrons by SF 6 takes place in a charge-polarisation potential of r −4 -distance dependence. A classical description of the orbital motion of electrons and SF 6 would lead to the Langevin capture rate constant k cap = 2π e (α/μ) 1/2 (e = electronic charge, μ ≈ electronic mass, α = polarisability of SF 6 ). However, the motion of the electron is not classical. Instead, s-wave quantum scattering governs the process. In addition, the process is of non-Born-Oppenheimer type with coupling between electron and nuclear motions such that the simple Langevin approach cannot be applied. Unlike the Langevin capture rate constant, k cap now depends both on the gas and the electron temperature, and even on gas pressure (see, e.g. Refs [8,9]). It needs collisions with other molecules in the bath gas to stabilise the vibrationally highly excited SF 6 − * . Alternatively, the stabilisation may be radiative, i.e. may involve IR emission. Figure 1 shows modelled pressure dependences of thermal attachment rate constants k at (T, P). If stabilisation does not take place, SF 6 − * either autodetaches the electron or fragments to SF 5 − + F. The latter process again is governed by a charge-polarisation potential (at least at long range). Statistical rate theory allows one to model the energy E-and angular momentum J-specific rate constants for electron detachment and ion fragmentation [10]. Figure 2 shows a comparison for J = 0. At low energies, autodetachment dominates, whereas dissociation takes over at energies above the dissociation energy. Obviously, the onset of the two processes is also determined by the value of the electron affinity of SF 6 given above.
The cross sections for dissociative electron attachment [11,12] to SF 6 have a peculiar shape such as illustrated in Figure 3. At low energies, the decrease reflects the decrease of the capture cross section with increasing electron energy E el . Here, the temperature-dependent high energy tail of the Figure 2. Specific rate constants for electron detachment (k det , SF 6 − → SF 6 + e − ) and dissociation (k dis , SF 6 − → SF 5 − + F) of SF 6 − * (modelling from Ref. [10]; the energies E should be decreased by 0.17 eV after re-evaluation [6] of E a ).  6 as a function of electron energy E el and gas temperature T gas (experimental points from Ref. [11]; modelling from Ref. [10], dashed curves: total cross sections). Boltzmann distribution of thermal SF 6 molecules (together with the electron energy) produces some SF 6 − * which is capable of dissociation. With increasing electron energy, E el finally exceeds the dissociation energy of SF 6 − * . Then the sum of E el and the thermal energy of SF 6 leads to increasing ion fragmentation, before finally the drop of the capture cross section leads to the marked decrease of the dissociative cross section at large E el . What looks like a 'resonance in the cross section' near E el = 500 meV has nothing to do with a resonance, but is the result of a competition of several rate processes and their convolution over energy distributions.
Similar to the dissociative cross sections, also the prod- ) have a peculiar shape such as shown in Figure 4. The decline of R at lower pressures in Figure 4 corresponds to collisional quenching of the excited SF 6 − * produced by the primary electron attachment, while the increase of R at higher pressures (for T = 620 K) is brought about by the thermal reactivation of SF 6 − after primary collisional quenching. Kinetic modelling allows one to understand the details of the shown R(T, P) (see Ref. [3]).
From quantum chemistry to dissociation kinetics: what we need to know in this case, first, obviously is the proper thermochemistry. Then, the non-Born-Oppenheimer electron attachment (and detachment) dynamics needs to be understood which is far from being the case today. Instead, empirical models still are necessary to bridge unknown territory (see Ref. [10]). Finally, the dissociation dynamics and collisional deactivation of highly excited molecular ions needs to be quantified. The latter has much in common with the analogous processes for neutral molecules  [13], experimental data given in Ref. [13]).
although the potential energy surfaces and the properties of collisional energy transfer are different. Nevertheless, such processes for ionic and neutral species should be treated on a common level. The following sections present examples for neutral reaction systems.

Dissociation/recombination studies in the CH 4 ↔ CH 3 + H and C 2 F 4 ↔ 2 CF 2 reaction systems
The temperature and pressure dependence of methane dissociation CH 4 ( + M) → CH 3 + H ( + M) and the reverse combination of CH 3 with H to form CH 4 have been investigated in remarkable detail and are particularly suitable for a demonstration of the present situation. Examples [13,14] [15][16][17] and, for more details, see [18][19][20]). The low-pressure behaviour reflects the properties of collisional energy transfer on CH 4 -M potential energy surfaces (see Refs [21,22]). As the two potentials now are known, one may perform a complete modelling of the reaction dynamics on these surfaces and study how close the experiment and theory come. At the same time, one may ask which details of the potentials matter most and which would be the minimum knowledge required for a meaningful comparison of experiment and theory.
We first consider the high-pressure limit of the CH 4 reaction system governed by the CH 4 potential only. Two properties of the potential turn out to be of primary im-  [13], experimental data given in Ref. [13]). portance: the minimum-energy path (MEP) potential and the anisotropy of the potential. In the CH 4 case, a Morsetype MEP potential fairly well reproduces the ab initio results (see Ref. [20]). Using the MEP potential only and treating the orbital motion of H around CH 3 leads to a capture rate constant (analogous to the Langevin rate constant in electron capture, see above) such as described by phase-space theory (PST) or loose activated complex theory. However, the presence of an anisotropy of the potential reduces the capture rate by introducing 'dynamical hindrance' (or 'tightening of the activated complexes'). One way to account for this effect is the use of a model potential. For example, a potential of the form was considered in Ref. [14] (with a fitted Morse parameter β, the dissociation energy D, the H-CH 3 distance r, the angle θ between the H-CH 3 line and the CH 3 plane, and an anisotropy amplitude factor C). Alternatively [23], one may employ a different decay parameter α in the second  [14], experimental data given in Ref. [14]).
term of Equation (3.1), e.g. here replace β by 2α and determine C from the corresponding deformation vibrations at r = r e . Ab initio potentials obviously differ from this simple form. Therefore, two possibilities to characterise an average overall anisotropy have been tested. First, the highpressure capture rate constant was calculated by classical trajectories both on the ab initio potential and on the model potential, which allowed for a fit of the ratio C/D (as done in Ref. [14]). Second, transitional-mode frequencies along the MEP of the ab initio potential were determined and trajectory calculations were performed on the corresponding model potential (such as described for the reaction 2CF 2 → C 2 F 4 below). Figure 8 illustrates the results for the CH 4 system. The upper points are for the isotropic potential, i.e. for Figure 8. High-pressure limiting rate constants for CH 4 → CH 3 + H (theoretical modelling from Ref. [14]; lines: modelling with potential of Equation (3.1) and C/D = 2.5, points: modelling with the ab initio potential of Refs [16,17]; upper line and points: isotropic potential, i.e. PST, lower line, and points: anisotropic potential).
PST, either from the Morse model potential or from the ab initio potential, the lower results are for the full anisotropic potential (points from the ab initio results and the line with the model potential of Equation (3.1) and a fitted ratio C/D = 2.5). One observes that the simple model potential works as well as the complete ab initio potential, confirming that the MEP potential and an average overall anisotropy parameters need to be known with first priority, while further details of the potential are less important for kinetic quantities like dissociation/association rates. The reduction of the capture rate constant by the anisotropy of the potential is well illustrated in Figure 8, PST overestimating the rate by about a factor of 2. We note that the analysis of the ab initio potential of Ref. [16] in Ref. [14] led to an effective Morse parameter β ≈ 2 Å −1 , while the analysis of the anisotropy of the ab initio potential in Ref. [19] gave α ≈ 0.7-0.8 Å −1 (being close to the value of 1 Å −1 suggested long ago in Ref. [23]). With a ratio α/β ≈ 0.4 this close to the 'normal' ratio of 0.5 derived from a series of typical dissociation and recombination reactions in Ref. [24]. The low-pressure limiting rate constant requires solution of the master equation for intermolecular collisional energy transfer [21,22,[25][26][27]. Assuming very efficient energy exchange, 'strong collision rate constants' are obtained as upper limits, only containing an overall collision number. The assumption that the latter can be identified with the Lennard-Jones collision number has been confirmed by classical trajectory calculations (see, e.g. Refs [21,22] for CH 4 collisions with inert colliders M). On the other hand, more efficient colliders like M = H 2 O, are characterised by larger collision numbers Z (see, e.g. Refs [22,28]). When collisional energy is less efficient, the rate constant is reduced by a 'collision efficiency' β c , being related to the average energy transferred per collision < E>. The contribution of vibrational and rotational energy transfer to < E> has been investigated in detail [21,25]. Again minor details do not matter too much, but an almost temperature independent < E> in practice generally has to be used as a fit parameter (a temperature independence of < E> corresponds to [25] an increase of the average energy transferred per down collision < E down > ∝ T 1/2 ). The necessity to use this fit parameter obviously presents a limitation to the prediction of low-pressure rate constants. For the methane system, this problem has been overcome. Here, the trajectory calculations of Ref. [21] on the ab initio potential for the CH 4 -M interaction led to < E> values in agreement with values derived by evaluating experimental low-pressure rate constants [28].
With the information on the dissociation/association and collisional energy transfer dynamics obtained from calculations on ab initio potentials and simplified versions of such potentials, one is in the position to model full falloff curves of the dissociation/association reactions [14,26]. The results of such modelling are included in Figures 5-7. Obviously kinetics averages out finer details of the Figure 9. Dependence of transitional mode frequencies on the C-C bond length r in C 2 F 4 (quantum-chemical calculations from Ref. [30], α/Å −1 = anisotropy parameter, see the text). potentials and of the dynamics and only few key properties need to be known. These are the parameters D, β, and C (or α/β) on the side of the intramolecular potential of Equation (3.1), and Z and < E> on the side of the collisional energy transfer.
The interplay between the attractive and anisotropic parts of the potential in dissociation/association dynamics has been demonstrated extensively with the statistical adiabatic channel model (SACM) [23] in combination with classical trajectory calculations (CT) [29]. In the original version [23] of the SACM, the anisotropy of the potential was expressed by the quantum states of deformation motions, changing from bending vibrations into free rotations along the MEP of dissociation. We illustrate this feature of an anisotropic potential for the C 2 F 4 → 2CF 2 dissociation. Figure 9 shows the results of quantum-chemical calculations [30] of vibrational frequencies of five torsional modes as a function of the C-C bond length r. One observes an exponential decay, confirming suggestions from the original SACM [23] and presenting the mentioned alternative to the anisotropy of the model potential of Equation (3.1).
Inserting the corresponding ratio of the decay parameter α ≈ 1.2 Å −1 and the effective Morse parameter β ≈ 5.2 Å −1 , i.e. α/β ≈ 0.23, into the modelling relationships from SACM/CT calculations [29], leads to high-pressure rate constants which are in close agreement with experimental data [30] (see Figure 10). As the calculation is based on ab initio potential data and does not involve additional empirical fit parameters, one concludes that the leading properties of the potential have been characterised realistically and are validated by the experimental results. In particular, the unusually small value of the ratio α/β (being much smaller than the normal value of this ratio such as observed for the CH 4 system described above) is identified to be responsible for the small value of the high-pressure association rate constant and its markedly positive temperature coefficient (see Figure 10). Figure 10. High-pressure limiting rate constants for 2 CF 2 → C 2 F 4 (line: SACM/CT calculations from Ref. [30] with α/β ≈ 0.23, experimental data given in Ref. [30]).

followed by HO 2 * + M → HO 2 + M
The recombination reaction H + O 2 → HO 2 plays a key role in the oxidation of hydrogen and hydrocarbons and, thus, is one of the most important reactions of combustion chemistry. Comparing its potential energy surface with that of H + CH 3 →CH 4 , one observes marked differences, mainly in the MEP potential such as illustrated in Figure 11. While the CH 4 -and C 2 F 4 -systems were characterised by simple Morse-type MEP potentials, now there is something like a 'shoulder' in the potential which may be the remnant of an avoided crossing of electronic states. This has profound consequences for the capture dynamics. Employing the ab initio potential from Ref. [31], adiabatic channel potential curves were constructed in Ref. [32]. Figure 12 shows examples of such curves, along which the quantum Figure 11. Minimum-energy path potential V(R) for HO 2 → H + O 2 (a 0 = 0.529 Å , from Ref. [32]; points and full line: ab initio potential of Ref. [31], dashed and dotted lines: simpler analytical potentials, see Ref. [32]).  Figure 13 shows thermal capture rate constants, in the upper curve for PST (neglecting the anisotropy of the potential) with clear evidence for transition state (TS) switching, from an outer TS at low temperatures to an inner TS at temperatures above about 50 K. Taking into account the anisotropy of the potential, the capture rate constant k cap decreases to the lower curve, but still showing the consequences of TS switching. Figure 13. Capture rate constants k cap for H + O 2 → HO 2 (from Ref. [32], dashed line: with isotropic long-range potential, upper full line: PST with MEP potential of Figure 11, lower full line: with anisotropic ab initio potential from Ref. [31]). In the H + O 2 system, we first encounter the 'openshell dilemma' of radical-radical reactions. All reaction partners are in open-electronic shells: H( 2 S 1/2 ), O 2 ( 3 g ), and HO 2 ( 2 A , 4 A ). The question arises which of the finestructure states contribute to the reaction. Simply assuming that only the lowest states with a thermal population f el (T) are relevant, conventionally one assumes that the highpressure recombination rate constant k rec,∞ here is given by with f el (T) = 1/3 (at 'not too low temperatures' and assuming that only doublets contribute). We come back to this point in the next section. Comparing this calculated k rec,∞ with experimental falloff curves from Ref. [28], one obtains very good agreement (see Figure 14). The agreement even reaches over into the liquid phase, where H + O 2 → HO 2 plays an important role in the radiation chemistry of water [33]. For the full falloff curves [28], one needs information on collisional energy transfer again such as described above for the CH 4 + M system. One should mention that M = H 2 O is a particularly efficient collider. It was suggested in Ref. [28] to identify the corresponding collision number Z with an HO 2 -H 2 O dipole-dipole capture rate constant instead of a Lennard-Jones collision number.

Open-electronic shell effects in radical-radical reactions
The question arises whether Equation (4.1) applies in general, i.e. whether a capture-controlled bimolecular reaction of open-electronic shell radicals proceeds on a single-potential energy surface, connecting the lowest electronic finestructure states of the reactants with the electronic ground Figure 15. Long-range potential energy curves for C + HO → CHO (from Ref. [34], numbers = values).
state of the intermediate adduct, and whether Equation (4.1) accounts for this. The fine-structure states at large distances generally are degenerate or nearly degenerate. When the reactants approach each other, then there may be a multitude of electronic states of the adduct, before the latter states sufficiently separate at close approach. Figure 15 illustrates such a situation [34] for the association process C + HO → CHO. During the capture kinetics, there will be efficient non-Born-Oppenheimer mixing of the shown states. In order to quantify this mixing, the potential has to be determined including long-range electrostatic, dispersion, induction, and exchange interactions; in addition spin-orbit and electronic-rotational coupling has to be taken into account. The reaction C + HO → CHO is the single example where the corresponding non-Born-Oppenheimer coupling dynamics has been implemented into an adiabatic channel treatment accounting for electronic-rotational (rotronic) couplings [35]. The results of this generalised SACM treatment using asymptotic potentials here were compared with detailed calculations based on Equation (4.1), see also Ref. [36]. Figure 16 compares the resulting association rate constants k ass . Over the range 15-200 K, the non-Born-Oppenheimer rate constant markedly exceeds the single-potential rate constant based on Equation (4.1) (k rec,∞ and k ass are equivalent). Apparently, rotronic coupling within the multitude of electronic states arising from the degenerate separated reactants generates a large amount of state mixing. This then leads to contributions from higher fine-structure states to capture into the electronic ground state of the adduct.
Other radical-radical reactions should be treated in a similar way. There is particular interest in the reaction HO + O → HO 2 → H + O 2 . Low temperature experimental rate constants are shown in Figure 17. The experimental rate differ considerably below about 150 K. One would hope that theory helps to settle the situation. It looks that full quantum time-independent calculations of cross sections and classical trajectory calculations from Ref. [37] favour the lower experimental values. However, these theoretical results were based on Equation (4.1) and did not account for rotronic couplings like those treated for C + HO in Ref. [35]. Figure 17 may suggest that the latter effects raise the   [40] compared with experimental points, for details see Ref. [40]; high temperature points from the reverse reaction converted with the equilibrium constant). rate constants to the upper values shown [38]. First steps towards a non-Born-Oppenheimer treatment of the reaction HO + O → HO 2 have been done in Ref. [39], but more work is required. Extending the temperature range of Figure 17 up to 5000 K, where data for the reverse reaction H + O 2 ⇔ HO 2 → HO + O are available, one reaches conditions where a single-potential Born-Oppenheimer treatment becomes sufficient. Figure 18 compares the detailed results from Ref. [40] with experiments. Here, from 300 to 5000 K, experiment and theory agree satisfactorily without that reaction parameters have to be empirically fitted. However, this agreement was only obtained after the enthalpy of formation of the OH radical finally could be established in Ref. [41]. Without proper thermochemistry such agreement could not have been reached.

Conclusions
From quantum-chemistry to dissociation dynamics: of course, a complete and accurate knowledge of the potential energy surface(s) of the reaction would be desirable to have and quantum-scattering calculations on the potential(s) should be made. However, this may involve enormous effort whose finer results would finally be lost by thermal averaging. Therefore, the simpler approaches described here may provide sufficient insight for the time being and a combination of the semiquantitative theoretical results with experimental data may lead to the presently most realistic values of the rate constants.