Mean-Potential Statistical Model and Role of Permutation Symmetry

Article pubs.acs.org/JPCA

Low-Energy H+ + H2 Reactive Collisions: Mean-Potential Statistical Model and Role of Permutation Symmetry Tasko P. Grozdanov*,† and Ronald McCarroll*,‡ †

Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia, and Laboratoire de Chimie Physique-Matière et Rayonnement (UMR 7614 du CNRS), Université Pierre et Marie Curie, 75231-Paris Cedex 05, France

‡

ABSTRACT: Statistical theory based on a mean isotropic potential deduced from a full potential energy surface is applied to the complex-forming, reactive H+ + H2 system in the interval of collision energies Ec = 10−3 to 0.5 eV. We present expressions for the reaction probabilities that incorporate the full permutation symmetry of the protons and compare our results with other statistical models and full quantum mechanical approaches that take account this symmetry correctly, approximately, or erroneously for the exchange rearrangement mechanism of the reaction.

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ple1,2). Various statistical models can be derived depending on the level of sophistication introduced in deriving the capture probabilities. The simplest are the Langevin-type classical models, obtained by assuming an isotropic interaction potential between the fragments which in combination with the centrifugal potential determines, for a given collision energy, the maximum orbital angular momentum (or impact parameter) for which a capture, namely, a complex formation, can occur. Mean isotropic potentials can be obtained either by angle-averaging of the anisotropic asymptotic interactions (for low collision energies) or by angle-averaging of (if available) the ab initio PES.4 These models can be improved by taking into account the quantum tunnelling through a potential barrier.5,6 The next level of sophistication is obtained by taking into account the anisotropy of the interaction explicitly. This can be achieved (at low collision energies) by employing adiabatic capture theories such as the statistical adiabatic channel model7 or the adiabatic capture with centrifugal sudden approximation.8 More recently, the capture probabilities have been determined by solving the close-coupled equations in the limited parts of configuration space but employing the ab initio PES (the so-called statistical quantum mechanical (SQM) model9). Interestingly, the same quality of the results has been obtained by using the capture probabilities obtained by running batches of classical trajectories on the limited part of the PES.10 Reaction 1 involves identical nuclei and therefore requires the incorporation into the theory of the quantum mechanical symmetrization postulate.11,12 In addition, by neglecting the

INTRODUCTION We consider the low-energy, complex-forming, reactive scattering process: H+ + H 2(v ,j) → (H3+)* → H 2(v′,j′) + H+

(1)

where (v,j) and (v′,j′) are the vibrational and rotational quantum numbers of the internal motion of the diatom in the reactant and product arrangements. The potential energy surface (PES) of H3+ has a deep (4.6 eV) potential well at small reactant (product) separations which can support a large number of bound states and resonances of the collision complex. As a consequence, the full quantum mechanical treatment of the above reaction involves a very large number of coupled channels and represents a challenging problem. However, if the lifetime of the complex is long enough to allow for randomization of the energy among the internal degrees of freedom, then the decay of the complex into the accessible product channels can be expected to proceed according to their statistical weights. The statistical description of the above reaction is then appropriate. The formulation of the statistical theory of reactive collisions originates from nuclear physics but its application to chemical reactions can be traced back to the works of Light1 and Miller.2 It is, however, only in recent years that it has been demonstrated that the results of statistical models compare favorably with quantum mechanical calculations when quantities averaged over the quantum oscillations are of interest (see, for example, the review by Gonzalez-Lezana3 for a number of atom−diatom reactive systems). It turns out that the only required input for the implementation of the statistical theory is a knowledge of the complex formation (or capture) probabilities from reactant and product channels (the decay probabilities being related to the formation probabilities through the microreversibility princi© 2012 American Chemical Society

Received: November 15, 2011 Revised: April 9, 2012 Published: April 9, 2012 4569

dx.doi.org/10.1021/jp210992g | J. Phys. Chem. A 2012, 116, 4569−4577

The Journal of Physical Chemistry A

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hyperfine interaction the conservation of the total nuclear spin should also be taken into account.12 Some time ago, in his work on ortho−para (that is, j = odd → j′ = even) conversion of H2 in reaction 1, Gerlich13 developed a statistical model that allows for the indistinguishability of the protons in an approximate way. At low energies, “capture” has been determined using classical trajectories on the correct polarization and quadrupole potentials, whereas at energies above 0.1 eV “complex formation” has been defined using “minimum exchanges” trajectories on a diatomic-inmolecule (DIM) PES.13 The range of collision energies covered in this work was Ec = 10−3 to 1 eV. More recently, reaction 1 has been studied by GonzalezLezana et al.14 in the context of a comparison between the classical trajectory based, statistical method (SQM) and the time independent quantum mechanical (TIQM) method. Their calculations have been performed using the global ARTPS -PES,15 which did not incorporate the long-range asymptotic interaction. However, the collision energy of Ec = 0.44 eV considered in their work is sufficiently high to justify the neglect of the long-range interaction. Finally, very recently, Honvault et al.16,17 have studied extensively reaction 1, comparing the SQM and TIQM methods over a wide range of low energies Ec = 10−6 to 0.1 eV and using the new VLABPPES18 with correct long-range asymptotic behavior. However, in our opinion, in all these works the formulas applied within the SQM method were appropriate for exchange reactions of the type A+ + B2 → AB + B+; that is, the indistinguishability of all of the nuclei has not been fully taken into account. Our goal in the present work is to study reaction 1 by using the mean-potential statistical (MPS) model, that is, the simplest, Langevin-type statistical method based on the isotropic mean potential4 derived from the above-mentioned two global PESs15,18 but with full implementation of the permutation symmetry,12 and to compare our results with those of Gerlich,13 Gonzalez-Lezana et al.14 and Honvault et al.16,17 In this way, we can test both the validity of the mean-potential method and the importance of the full implementation of the symmetrization postulate. Recently, the MPS model has been applied to the H+ + D2 → HD + D+ reaction.19 It is important to assess the reliability of the simplest statistical methods in cases like reaction 1, for which full quantum mechanical calculations are feasible. In more complex reactions involving, for example, polyatomic molecules these could be the only possible realistic methods that might be applied.20

Pvj ,v ′ j ′(E ,J ,Π) ⎧ 2 (1) for j odd and j′ odd ⎪ Pvj ,v ′ j ′(E ,J ,Π) ⎪3 1 ⎪ + Pvj(2),v ′ j ′(E ,J ,Π) ⎪ 3 =⎨ ⎪ 1 P(2) (E ,J ,Π) for j odd and j′ even ⎪ 3 vj ,v ′ j ′ ⎪ ⎪ Pvj(2),v ′ j ′(E ,J ,Π) otherwise ⎩

where Pvj(1),v ′ j ′(E ,J ,Π) =

Pvj(2),v ′ j ′(E ,J ,Π) =

pvj (E ,J ,Π) pv ′ j ′ (E ,J ,Π) ∑v ″ j ″= odd pv ″ j ″ (E ,J ,Π)

∑v ″ j ″ pv ″ j ″ (E ,J ,Π)

(5)

and the summations in the denominators are restricted to states that are energetically accessible. The cumulative complex formation probabilities (CCFPs) pvj(E,J,Π), as discussed in the Introduction, can be calculated in a number of different ways depending on the method used to describe the collision dynamics. The general form of eqs 3−5 is, however, independent of that and is the consequence of the indistinguishability of the protons in the quantum mechanical treatment of reaction 1 and the assumption of the conservation of the total nuclear spin during the collision. It can be obtained by combining the results of the quantum mechanical treatment of the atom−diatom rearrangement collisions with three identical nuclei by Miller21 and his formulation of the statistical theory of chemical reactions.2 It can also be directly derived from the generalized formulation of the statistical scattering theory by Park and Light12 (see Table 2 and eqs 22−29 in this reference). The superscripts (1) and (2) in eqs 3−5 correspond respectively, to irreducible representations A1 and E of the complete nuclear permutation group S3.12 In our approach, we calculate the CCFPs pvj(E,J,Π), by summing over all possible quantum numbers l of relative orbital angular momenta: J+j

pvj (E ,J ,Π) =

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THEORETICAL METHOD We start with the formula for the state-to-state integral cross section for eq 1:

∑

δ Π,(−1) j+lp(l ,Ec) (6)

where the Kronecker delta stands for the conservation of parity, and the elementary capture probabilities p(l,Ec) are given by ⎧ l ≤ lm(Ec) ⎪1 p(l ,Ec) = ⎨ ⎪ ⎩ 0 l > lm(Ec)

∑ (2J + 1)Pvj ,v ′ j ′(E ,J ,Π) J ,Π

(4)

pvj (E ,J ,Π) pv ′ j ′ (E ,J ,Π)

l =|J − j|

πℏ2 σvj ,v ′ j ′(E) = (2j + 1)2μEc

(3)

(2)

(7)

The above expression corresponds to a simple classical Langevin model assuming that capture (namely, complex formation) occurs whenever the collision energy is larger than the maximum of an effective isotropic potential. Therefore, the lm(Ec) in eq 7 is defined as the maximum integer l that satisfies

where J is the total motional angular momentum quantum number, Π = ±1 is the total parity quantum number, and μ ≃ 2mp/3 is the reduced mass, where mp is the proton mass. The total energy is E = Ec + Evj = E′c + Ev′j′, where Ec(E′c) and Evj(Ev′j′) are respectively the collision and internal rovibrational energies of the reactants (products). The state-to-state partial cumulative reaction probability (CRP), within the microcanonical statistical method, is given by

⎧ ℏ2l(l + 1) ⎫ ⎬ + Ec > maxR⎨ V ( R ) ̅ ⎭ ⎩ 2μR2 4570

(8)



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case functions merge at higher energies. For relative orbital angular momentum quantum numbers l > 47 the maximum (centrifugal barrier) in the effective potential on the right-hand side of eq 8 disappears (for both ARTPS-PES and VLABPPES), so that we have lm(Ec) = 47 for Ec > 0.82 eV. But these relatively high energies are out of the scope of the present study. The “collision” (or “dynamic”) information contained in Figure 1b is actually the only one needed to calculate, within the framework of the MPS model, all cross sections, probabilities, and distributions of interest with respect to reaction 1.

In our calculations we have used as an isotropic interaction potential V̅ (R), the mean potential obtained from angle averaging:4 V ̅ (R ) =

1 2

∫0

π

V (R ,r =re ,θ) sin θ dθ

(9)

+

where V(R,r,θ) is a PES of the H3 molecule, R is the distance between the H+ and the center of mass of H2, r is the bond distance of H2 (re = 1.40104 au is its equilibrium value), and θ is the angle between the vectors R⃗ and r.⃗ The zero of the potential is taken at R → ∞. Figure 1a shows mean potentials V̅ (R) as well as two extreme cuts V(R,r=re,θ=0) and V(R,r=re,θ=π/2) corresponding to both

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RESULTS Intrinsic Rate Coefficients and Cross Sections. Previously, Gerlich13 has considered the ortho−para transitions in H2 in reaction 1 by using a statistical model that approximately, relying on the results of Quack,11 took into account the permutation symmetry of protons and conservation of total nuclear spin. The corresponding state-to-state partial CRP has been taken in the form13 PvjG,v ′ j ′(E ,J ,Π) =

pvj (E ,J ,Π)gjj ′pv ′ j ′ (E ,J ,Π) ∑v ″ j ″ gjj ″pv ″ j ″ (E ,J ,Π)

(10)

with ⎧ 5 for j odd and j′ odd gjj ′ = ⎨ ⎩1 otherwise

(11)

The CCFPs pvj(E,J,Π) have been obtained from empirical analytic formulas established by combining the classical trajectory calculations using the correct long-range potential for low-energy collisions and a DIM PES for higher energies.13 The dependence on the CCFPs in eqs 3−5 and eqs 10 and 11 is different only if an initial ortho-H2 (reactant with odd j) is involved, although even in these cases at higher energies (when subsums over even and odd j″ become practically the same) the two expressions merge. Actually, it is easy to see that in this limit, CRPs that correspond to ortho → ortho, ortho → para, para → ortho, and para → para transitions are all proportional to P(2) vj,v′j′(E,J,Π) with respective prefactors: 5/3, 1/3, 1, and 1. The latter are exactly the “average statistical correction factors”, derived by Quack.11 We first compare the results of our MPS model with those of Gerlich13 for the intrinsic rate coefficients, which are simply related to state-to-state reaction cross sections:

Figure 1. (a) Mean potentials V̅ (R) and potential cuts for collinear V(R,r=re,θ=0) and perpendicular V(R,r=re,θ=π/2) configurations corresponding to ARTPS-PES (thin lines) and VLABP-PES (thick lines). (b) Maximal possible orbital angular momentum quantum number lm leading to complex formation as a function of the collision energy Ec for ARTPS-PES (thin lines) and VLABP-PES (thick lines).

ARTPS-PES and VLABP-PES. Although both mean potentials merge at short distances where a deep (−4.6 eV) potential minimum is located, at large distances only VLABP-PES possesses the correct asymptotic behavior, due to the built-in multipole long-range interactions.18 Because for the description of the low-energy collisions the correct long-range behavior is essential, we shall almost exclusively use the mean-potential corresponding to VLABP-PES, except in a few cases when it will be explicitly noted. We note also that in the case of the VLABP-PES there is a broad barrier in the collinear (θ = 0) configuration, which, however, disappears after averaging over angles. At very low collision energies this can lead to discrepancies with other methods that use full PES to determine the reaction probabilities. Knowing the mean potentials V̅ (R) and using eq 8 it is easy to determine the lm(Ec) functions, which are shown in Figure 1b for ARTPS-PES and VLABP-PES. The largest differences are at extremely low collision energies whereas the two stair-

kvj ,v ′ j ′(E) = (2Ec /μ)1/2 σvj ,v ′ j ′(E)

(12)

Figure 2 shows a comparison for the endothermic reactions corresponding to excitation of a single rotational quanta from the ground vibrational state. The nonsmooth (zigzag) behavior of our results is the consequence of the step-function used in eq 7. For initial j = 0, 2 the agreement of our results with those of Gerlich is very good, except that the latter tend to somewhat higher values at collision energies Ec > 0.2 eV. As seen below, the same behavior is found in all other cases considered. The reason for this discrepancy could be due to the fact that Gerlich used one type of empirical formulas for capture probabilities in the region Ec < 0.22 eV, namely those based on calculations using long-range asymptotic interactions, whereas for Ec > 0.22 eV he used other formulas established by using a DIM PES. So the discrepancy could be due to different PESs used in the two 4571



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all subsequently derived data in the figures to follow) are of limited precision because they have been taken directly from the published text. Recently, Honvault et al.16,17 studied reaction 1 on the wide range of low energies Ec = 10−6 to 0.1 eV, by employing the SQM and TIQM methods and the VLABP-PES. Though both methods require solutions of coupled-states quantum mechanical equations, the SQM method employs solutions in the limited regions of PES to define channel-dependent capture probabilities and apply them in the corresponding statistical model. The TIQM method requires a full quantum mechanical solution and, assuming the convergence requirements are fulfilled, provides, in principle, “exact” results. The SQM and TIQM state-to-state integral cross sections are compared in Figure 4 with our MPS and Gerlich’s (where

Figure 2. State specific intrinsic rate coefficients for endothermic j → j + 1 reactions as a function of collision energy. Thick solid lines are present MPS results, thin solid lines are present MPS results but calculated using eq 10 (with MPS CCFPs), and dashed lines are the results of Gerlich.13

calculations, although the methods of determining the capture probabilities are also different. In the cases of initial ortho-H2, as anticipated above, there are differences also at low energies. This is particularly seen in the j = 1 case in Figure 2. However, calculations using Gerlich’s formulas eqs 10 and 11 but our CCFPs eqs 6−8, show much better agreement with Gerlich’s results at low energies. Our two sets of MPS results merge at higher energies. In the j = 3 case, with a higher threshold energy, the differences between the three sets of results are smaller. Similar trends are found in the cases of exothermic reactions leading to de-excitation by the single rotational quanta, as is shown in Figure 3. Again, our CRP formulas differ from Gerlich’s formulas only if ortho-H2 reactants are involved. However, it is difficult to draw any specific conclusions except that overall agreement with Gerlich’s results is fairly good. We note also that Gerlich’s results shown in Figures 2 and 3 (and

Figure 4. MPS (thick solid lines), Gerlich’s13 (dashed lines), TIQM16 (thin solid lines), and SQM16 (dash-dot lines) state-to-state integral cross sections for the reaction H+ + H2(v=0,j) → H2(v′=0,j′) + H+ with initial (a) j = 1 and (b) j = 0.

available and converted from Figures 2 and 3) results. In all cases, H2 molecules are initially and finally in ground vibrational states. Figure 4a shows rotational transitions from the first excited state j = 1 to j′ = 0, 2, and 3. As we may expect from the preceding paragraph concerning the intrinsic rate coefficients, the MPS and Gerlich’s integral cross sections are in close agreement. The TIQM results exhibit a highly oscillatory and irregular structure, located between our MPS (and Gerlich’s) results that are higher and the SQM results that are lower. The irregular oscillations of the TIQM results are presumably due to the large number of scattering resonances, localized in the complex formation region (that is above the deep potential well), which take part in the reactive scattering process. It is interesting to note that the largest differences between MPS, TIQM, and SQM results appear for the only ortho−ortho, j = 1 to j′ = 3 transition. Figure 4b shows cross sections for reactions starting from the ground state j = 0 to j′ = 1, 2, and 3. One can see that the same ordering in magnitudes of the cross sections obtained by different methods persists in these cases too.

Figure 3. Same as Figure 2, but for exothermic j → j − 1 reactions. 4572



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The most striking fact of the comparisons in Figure 4 is the unexpectedly large difference between the results of statistical methods. The SQM predictions are considerably lower than those of the MPS and Gerlich models. Honvault et al.16,17 do not give explicit expressions for the partial CRPs which correspond to our eqs 4 and 5. However, from the text in Honvault et al.17 it seems that these authors have been using instead of our eq 5 the expression: ̃ ,v ′ j ′(E ,J ,Π) = Pvj(2)

pvj (E ,J ,Π) pv ′ j ′ (E ,J ,Π) ∑ ′vj̃ ̃ pvj̃ ̃ (E ,J ,Π) + ∑v∼′∼j ′ pv ̃′ j ̃ ′ (E ,J ,Π) (13)

where j and j ̃ are of the same parity. The above expression, however, is related to exchange reactions of the form A+ + B2 → B+ + AB and does not fully incorporate the permutation symmetry of all three protons in reaction 1. To clarify the relationship between eqs 5 and 13, it is useful to adopt the highenergy approximation where the subsums over even and odd “j”s in the denominators of eqs 5 and 13 are assumed to be equal, so that one finds ̃ ,v ′ j ′(E ,J ,Π) ≈ Pvj(2)

2 (2) Pvj ,v ′ j ′(E ,J ,Π) 3

(14)

Indeed, in many comparisons (especially at higher energies) in the following subsections the SQM results are roughly twothirds of the MPS results. The additional differences are, of course, due to the fact that CCFPs pvj(E,J,Π) are computed by two different methods. To check for this hypothesis, we have performed the MPS (that is employing our capture probabilities) cross-section calculations but using eq 13 instead of eq 5. A better scale for comparing thus obtained results with previous ones, is provided by the total CRP, which we consider in the next subsection. Total Cumulative Reaction Probabilities. Total state-tostate CRP is closely related to state-to-state integral cross section: Nvj ,v ′ j ′(E) =

Figure 5. MPS (thick solid lines, upper curves using eq 5 and lower curves using eq 13), Gerlich’s13 (dashed lines), TIQM16 (thin solid lines), and SQM16 (dash-dot line) state-to-state total CRP for the reaction H+ + H2(v=0,j) → H2(v′=0,j′) + H+. (a) j = 0 → j′ = 1 transition. Individual symbols at Ec = 0.44 eV correspond to MPS (squares), SQM14 (triangle), and TIQM14 (circle). (b) j = 1 → j′ = 0 transition.

⎤1/2 ⎡ 8 ⎥ kvj ,v ′ j ′(T ) = ⎢ ⎣ πμ(kBT )3 ⎦

σvj ,v ′ j ′(Ec)e−Ec / kBT Ec dEc

where T is the temperature and kB is the Bolzmann constant. In Table 1 we compare our MPS thermal rate coefficients for the two dominant low energy transitions j = 1 → j′ = 0 and j = 0 → j′ = 1 with Gerlich’s13 and the TIQM and SQM results of Honvault et al.16,17 It should be noted that the originally published TIQM and SQM rate coefficients of Honvault et al.16,17 are in error22 by a factor of (0.52918)2 representing the conversion between the atomic units and Å2. Data shown in Table 1 are corrected data. As can be seen, the MPS rates (just like the cross sections) for j = 1 → j′ = 0 and j = 0 → j′ = 1 transitions are somewhat lower than those of Gerlich, but higher than TIQM and SQM results. It is also noticeable that in the case of the j = 1 → j′ = 0 transitions, MPS and Gerlich’s rates slightly decrease while TIQM and SQM rates slightly increase with temperature. Partial Cumulative Reaction Probabilities and Rotational Distributions. First we study, for the range of fixed low collision energies, the J-dependence of the state-to-state partial CRP, defined as

J ,Π

(2j + 1)2μEc πℏ2

∞

(16)

∑ (2J + 1)Pvj ,v ′ j ′(E ,J ,Π)

= σvj ,v ′ j ′(E)

∫0

(15)

Panels a and b of Figure 5 show this quantity, respectively, for the transitions from j = 0 to j′ = 1 and j = 1 to j′ = 0. In addition to the MPS, Gerlich, TIQM, and SQM results, which correspond to those given in Figure 4, we have also shown MPS results obtained by using eq 13 instead of eq 5 (lower solid curves). It is obvious that these results are in good agreement with SQM predictions, thus supporting our hypothesis that Honvault et al.16,17 used eq 13 in their calculations. Also shown in Figure 5a, at Ec = 0.44 eV are the individual SQM and TIQM data recalculated from Gonzalez-Lezana et al.14 Although here the SQM point lies between the two MPS curves, the apparently low value of TIQM is puzzling. We shall return to calculations of Gonzalez-Lezana et al.14 in the following subsections. Thermal Rate Coefficients. Once the integral cross sections are known, the state-to-state thermal rate coefficients can be calculated from

Pvj ,v ′ j ′(J ,E) =

∑ Π=±1

Pvj ,v ′ j ′(E ,J ,Π) (17)

where Pvj,v′j′(E,J,Π) is given in eqs 3−5. We note that the “opacity functions” or “reaction probabilities” are usually defined as Pvj,v′j′(J,E)/(2j + 1). 4573



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Table 1. Thermal Rate Coefficients (cm3/s) for the Reaction H+ + H2(v=0,j) → H2(v′=0,j′) + H+ in Terms of the Temperature T (K)a T

MPS

10 20 30 40 50 60 70 80 90 100

2.196(−10) 2.142(−10) 2.119(−10) 2.107(−10) 2.099(−10) 2.092(−10) 2.086(−10) 2.077(−10) 2.068(−10) 2.057(−10)

10 20 30 40 50 60 70 80 90 100

7.838(−17) 3.839(−13) 6.504(−12) 2.676(−11) 6.251(−11) 1.100(−10) 1.644(−10) 2.221(−10) 2.801(−10) 3.368(−10)

Gerlich13 j = 1, j′ = 0 2.972(−10) 2.453(−10) 2.300(−10) 2.228(−10) 2.186(−10) 2.158(−10) 2.138(−10) 2.123(−10) 2.112(−10) 2.103(−10) j = 0, j′ = 1 1.084(−16) 4.611(−13) 7.470(−12) 3.007(−11) 6.933(−11) 1.210(−10) 1.801(−10) 2.426(−10) 3.062(−10) 3.687(−10)

TIQM17

SQM17

1.163(−10) 1.299(−10) 1.367(−10) 1.419(−10) 1.460(−10) 1.493(−10) 1.517(−10) 1.534(−10) 1.545(−10) 1.552(−10)

9.232(−9) 1.064(−10) 1.144(−10) 1.193(−10) 1.226(−10) 1.249(−10) 1.265(−10) 1.276(−10) 1.284(−10) 1.288(−10)

3.290(−17) 2.044(−13) 3.848(−12) 1.694(−11) 4.147(−11) 7.552(−11) 1.159(−10) 1.596(−10) 2.042(−10) 2.484(−10)

3.100(−17) 1.924(−13) 3.666(−12) 1.618(−11) 3.954(−11) 7.177(−11) 1.098(−10) 1.507(−10) 1.925(−10) 2.366(−10)

using short-range ARTPS-PES and long-range VLABP-PES), as well as the results of TIQM17 and SQM17 methods. At this very low collision energy, as seen from Figure 1b, one can expect a pronounced difference between the two sets of the MPS results, which is indeed confirmed in Figure 6a. The MPS probabilities calculated with the VLABP-PES are the largest and predict the largest Jm (maximal total angular momentum) that contributes to the reaction. The TIQM results exhibit the oscillatory structure, whereas the SQM results are the smallest and smooth. It should be noted that at this (and lower) collision energy the applicability of the statistical models can be questioned: the number of open channels may be too small, as well as the density of the scattering resonances with lifetimes long enough to allow for the statistical mixing. In the present version of the MPS model the simple classical overbarrier capture model may also cease to be applicable. In any case, the more systematical and extensive comparisons with TIQM results at extremely low collision energies are required to precisely define the low-energy limit of the applicability of the statistical models. At collision energy Ec = 10−3 eV, as seen in Figure 6b, the difference between the MPS results corresponding to two different PESs has diminished, the SQM data grow in magnitude and the oscillations of the TIQM probabilities encompass the MPS and SQM results. Results for the next considered collision energy Ec = 0.05 eV, are shown in Figure 6c. At his energy there is no difference between the two MPS calculations using the two PESs. The SQM and MPS results start to acquire similar shapes, the SQM data being about 2/3 of the MPS data. The TIQM results oscillate with somewhat smaller amplitudes and with an overall mean value of the probability which is located between the MPS and SQM data.

a

The TIQM and SQM results from Honvault et al.17 are corrected by factor of (0.52918)2 due to an error in original data.22

Figure 6a shows the J-dependence of P01,00(E,J) at collision energy Ec = 10−4 eV, as it is predicted by the MPS model (when

Figure 6. Partial CRP as a function of total angular momentum J for the H+ + H2(v=0,j=1) → H2(v′=0,j′=0) + H+ reaction at (a) Ec = 10−4 eV, (b) Ec = 10−3 eV, (c) Ec = 0.05 eV, and (d) Ec = 0.1 eV: diamonds, MPS results using ARTP-PES; squares, MPS results using VLABP-PES; triangles, SQM17 results; circles, TIQM17 results. 4574



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Figure 7. Same as Figure 6 but for the reaction H+ + H2(v=0,j=0) → H2(v′=0,j′=1) + H+ at (a) Ec = 0.02 eV, (b) Ec = 0.05 eV, (c) Ec = 0.08 eV, and (d) Ec = 0.1 eV.

et al.17 do not provide results for elastic transitions j = j′. The MPS and Gerlich13 data shown in Figure 8 correspond to contributions to elastic cross sections that originate from the complex formation mechanism.2

Notice, also, that Jm of the MPS results is smaller (by a unit or two) than that for SQM and TIQM calculations, which is also typical for all higher collision energies. In addition, though the MPS probability abruptly falls to zero for J > Jm, the decline of SQM and TIQM data is more gradual. This is the consequence of the absence of the angular dependence in the interaction potential and the employment of the simple Langevin capture model in the MPS method. All this applies also to the case of collision energy Ec = 0.1 eV, shown in Figure 6c. The only exception is that again small differences appear between the MPS results corresponding to two PESs. This can happen for two reasons: first, the two corresponding mean potentials are not exactly equal at small reactant separations, and second, the threshold regions of the opening of the new channels involve two very slowly departing fragments and the differences shown in Figure 1b at low energies can play a role in determination of the probabilities. Figure 7a−c shows the J-dependence of P00,01 (E,J), corresponding to the endothermic j = 0 → j′ = 1 reaction at collision energies Ec = 0.02, 0.05, 0.08, and 0.1 eV, respectively. All of the comments made above with respect to Figure 6c,d apply here to all shown cases. In particular, it is more obvious that results of the SQM and MPS calculations differ roughly by a factor of 2/3. To conclude the considerations of these low-energy collisions, we show in Figure 8 the distributions of the integral cross sections over the final rotational quantum numbers at Ec = 0.1 eV. As seen from Figure 8a, for initial j = 0 state the MPS, Gerlich13 and TIQM17 results are all in good agreement. The SQM17 results are for about a factor of 2/3 lower. For initial j = 1 state, as seen from Figure 8b, the situation is more complicated. Altough for final j′ = 0 and 2, the spread of data is small, for j′ = 3 the dispersion is the largest, as it is also at other collision energies (Figure 4). We note also that Honvault

Figure 8. Integral cross sections as a function of final-state rotational quantum number for the reaction H+ + H2(v=0,j) → H2(v′=0,j′) + H+ at Ec = 0.1 eV and for (a) j = 0 and (b) j = 1: squares, MPS results; diamonds, Gerlich’s13 results; triangles, SQM17 results; circles, TIQM17 results. 4575



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Turning now to comparisons with some earlier results of Gonzalez-Lezana et al.14 at somewhat higher collision energy for the reaction H+ + H2(v = 0,j = 0) → H2 + H+, we consider the partial CRP summed over the final states: Pvj(J ,E) =

∑ Π,v ′ j ′

Pvj ,v ′ j ′(E ,J ,Π) (18)

Figure 9 shows the J dependence of the P00(J,E) at collision energies Ec = 0.44 eV, as predicted by various methods of

Figure 10. Integral cross section distributions over the final rotational quantum numbers at collision energy Ec = 0.44 eV: squares, MPS results obtained using eq 5; triangles, SQM14 results; circles, TIQM14 results; diamonds, MPS results obtained using eq 13; stars, MPS results obtained using eq 21.

agreement with MPS (using eq 5) results whereas the SQM data are too low. Rearrangement Mechanisms. In the reaction of a proton with the hydrogen diatom, if protons are formally treated as distinguishable, it is possible to consider two rearrangement mechanisms12 of protons:

Figure 9. Partial CRP summed over the final states as a function of total motional angular momentum quantum number J, at collision energy Ec = 0.44 eV: squares, MPS results obtained using eq 5; triangles, SQM14 results; circles, TIQM;14 diamonds, MPS results obtained using eq 13 or eq 21.

H+ + H̃ 2 → H+ + H̃ 2

calculation. We note also that here all results have been obtained using short-range ARTPS-PES. This is, however, irrelevant because at this relatively high collision energy, as we have checked, the difference with results obtained when using long-range VLABP-PES is negligible. In the case of the MPS results, the summation in eq 18 can be performed analytically to obtain P00(J,E) = p(l=J,Ec), where the elementary capture probability (step function) is defined in eq 7 with lm = Jm = 38 for Ec = 0.44 eV. This dependence is shown in Figure 9 as the uppermost straight line formed by squares. The SQM14 results are also almost constant but grouped around the 2/3 and TIQM14 results oscillate around the SQM data. This indicates that in the paper by Gonzalez-Lezana et al.14 besides the SQM calculations, the TIQM calculations have also been performed without the full inclusion of the permutation symmetry of the protons, so that they are appropriate for a reaction of the type A+ + B2 → B+ + AB. This is further supported by the fact that if the MPS calculations are performed using eq 13 instead of eq 5, then, as seen from Figure 9, the agreement with the SQM results is excellent. For the same process, the distributions of the integral cross sections over the final rotational quantum numbers are shown in Figure 10. Again, the SQM and MPS (using eq 13) results mutually agree well and are close to TIQM results (apart from the oscillations of the latter at low values of j′) and they are all about factor of 2/3 lower than the MPS (using eq 5) results. The four points from Figure 10 corresponding to j′ = 1 are those also shown in Figure 5a. Now it is clear that the TIQM point in Figure 5a at Ec = 0.44 eV is too low because of the absence of the proper inclusion of full permutation symmetry. We note that at Ec = 0.1 eV, as seen from Figure 8a, we have a different situation: the TIQM results (which include full permutation symmetry of the protons) are in a very good

identity

̃ → H̃ + + HH exchange

(19)

In the formulation of our MPS model, in the Gerlich approach,13 as well as in the TIQM method used in Honvault et al.,16,17 the total contribution of both mechanisms has been included, as it should be due to the indistinguishability of the protons. On the other hand, as already discussed in the previous subsections, it seems that the SQM method employed in Honvault et al.16,17 and the SQM and TIQM methods in Gonzalez-Lezana et al.14 correspond to the description of the exchange mechanism only. However, using the “spin modification probability” method of Park and Light12 (see Table 2 and eq 35 in this reference), it is possible to derive the precise expressions for the partial CRPs corresponding to each of the rearrangement mechanisms. For the identity mechanism one finds Pvj(id) ,v ′ j ′(E ,J ,Π) ⎧ 2 (1) for j odd and j′ odd ⎪ Pvj ,v ′ j ′(E ,J ,Π) ⎪9 2 (2) ⎪ ⎪ + 9 Pvj ,v ′ j ′(E ,J ,Π) =⎨ ⎪ 2 (2) for j even and j′ even ⎪ Pvj ,v ′ j ′(E ,J ,Π) ⎪3 ⎪ ⎩0 otherwise

(20)

whereas for the exchange mechanism one finds 4576



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of course, that they proceed via the complex-forming mechanism. Additional studies and comparisons with quantum mechanical methods are also needed to establish the lower and upper energy limits of the applicability of various variants of the statistical models. For example, the recent studies of Jambrina et al.23,24 on H+ + D2 and D+ + H2 reactive systems indicate the possible departures from the “complex-forming” mechanism at higher energies. The problem is, of course, that the quantum mechanical calculations become prohibitively expensive with the increase of collision energy.

Pvj(ex) ,v ′ j ′(E ,J ,Π) ⎧ 4 (1) for j odd and j′ odd ⎪ Pvj ,v ′ j ′(E ,J ,Π) ⎪9 1 ⎪ + Pvj(2),v ′ j ′(E ,J ,Π) ⎪ 9 =⎨ ⎪ Pvj(2),v ′ j ′(E ,J ,Π) for j even and j′ odd ⎪ ⎪ 1 (2) otherwise ⎪ Pvj ,v ′ j ′(E ,J ,Π) ⎩3

(21)

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Equations 4 and eq 5 still hold. Of course, by summing eqs 20 and 21, we arrive at eq 3. If we use now our MPS model with eq 21 instead of eq 3, the calculated rotational distribution of the integral cross sections at Ec = 0.44 eV, for the exchange process ̃ H+ + H̃ 2(v=0,j=0) → H̃ + + HH

AUTHOR INFORMATION

Corresponding Author

*E-mail: T.P.G., [email protected]; R.M., ronald.mac_carroll@ upmc.fr. Notes

The authors declare no competing financial interest.

(22)

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is represented in Figure 10 by stars. As seen from Figure 10, these results oscillate around the predictions of other methods related to the exchange process. On the other hand, if the partial CRP summed over the final states, given by eq 18, is calculated using MPS and eq 21, the result is indistinguishable on the scale of Figure 9 from that obtained by using MPS and eq 13 (represented by diamonds). It is not surprising then that all methods related to the exchange process predict similar values of the total (summed over j′’s) cross sections at Ec = 0.44 eV: MPS using eq 21, σ00 = 22.42 Å2; MPS using eq 13, σ00 = 22.46 Å2; SQM,14 σ00 = 23.70 Å2 and TIQM,14σ00 = 23.13 Å2. The cross section that includes both contributions from the identity and the exchange mechanism, obtained by the MPS model using eqs 3 to 5, i.e., the correct value, is σ00 = 33.80 Å2.

ACKNOWLEDGMENTS We are grateful to P. Honvault and T. Gonzalez-Lezana for sending us the files with their results and for discussions concerning their previous work on the subject. T.P.G. acknowledges support by Ministry of Education and Science of the Republic of Serbia through the project No. 171020.

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REFERENCES

(1) Light, J. C. Discuss. Faraday Soc. 1967, 44, 14−29. (2) Miller, W. H. J. Chem. Phys. 1970, 52, 543−551. (3) Gonzalez-Lezana, T. Int. Rev. Phys. Chem. 2007, 26, 29−91. (4) Larrégaray, P.; Bonnet, L.; Rayez, J.-C. J. Chem. Phys. 2007, 127, 084308. (5) Larrégaray, P.; Bonnet, L.; Rayez, J. C. Phys. Chem. Chem. Phys. 2002, 4, 1571−1576. (6) Park, K.; Light, J. C. J. Chem. Phys. 2007, 126, 044305. (7) Smith, S. C.; Troe, J. J. Chem. Phys. 1992, 97, 5451−5464. (8) Clary, D. C. J. Chem. Soc., Faraday Trans.2 1987, 83, 139−148. (9) Rackham, E. J.; Huarte-Larranaga, F.; Manolopoulos, D. E. Chem. Phys. Lett. 2001, 343, 356−364. (10) Aoiz, F. J.; Rabanos, V. S.; Gonzalez-Lezana, T.; Manolopoulos, D. E. J. Chem. Phys. 2007, 126, 1616101. (11) Quack, M. Mol. Phys. 1977, 34, 477−504. (12) Park, K.; Light, J. C. J. Chem. Phys. 2007, 127, 224101. (13) Gerlich, D. J. Chem. Phys. 1990, 92, 2377−2388. (14) Gonzalez-Lezana, T.; Roncero, O.; Honvault, P.; Launay, J.-M.; Bulut, N.; Aoiz, F. J.; Banares, L. J. Chem. Phys. 2006, 125, 094314. (15) Aguado, A.; Roncero, O.; Tablero, C.; Sanz, C.; Paniagua, M. J. Chem. Phys. 2000, 112, 1240−1254. (16) Honvault, P.; Jorfi, M.; Gonzalez-Lezana, T.; Faure, A.; Pagani, L. Phys. Rev. Lett. 2011, 107, 023201. (17) Honvault, P.; Jorfi, M.; Gonzalez-Lezana, T.; Faure, A.; Pagani, L. Phys. Chem. Chem. Phys. 2011, 13, 19089−19100. (18) Velilla, L.; Lepetit, B.; Aguado, A.; Beswick, J. A.; Paniagua, M. J. Chem. Phys. 2008, 129, 084307. (19) Grozdanov, T. P.; McCarroll, R. J. Phys. Chem. A 2011, 115, 6872−6877. (20) Hugo, E.; Asvany, O.; Schlemmer, S. J. Chem. Phys. 2009, 130, 164302. (21) Miller, W. H. J. Chem. Phys. 1969, 50, 407−418. (22) Honvault, P.; Jorfi, M.; Gonzalez-Lezana, T.; Faure, A.; Pagani, L. Phys. Rev. Lett. 2012, 108, 109903(E). (23) Jambrina, P. G.; Aoiz, F. J.; Bulut, N.; Smith, S. C.; Balint-Kurti, G. G.; Hankel, M. Phys. Chem. Chem. Phys. 2010, 12, 1102−1115. (24) Jambrina, P. G.; Alvarino, J. M.; Aoiz, F. J.; Herrero, V. J.; SáezRábanos, V. Phys. Chem. Chem. Phys. 2010, 12, 12591−12603.

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CONCLUSIONS In the present study of reaction 1 we have shown that, by using our MPS model, fair agreement is obtained with previous statistical calculations of Gerlich13 for both the intrinsic rate coefficients and the final-rotational-state distributions. The use of expressions that exactly account for the permutation symmetry as opposed to approximate expressions of Gerlich,13 introduces only small differences at low collision energies for reaction involving initial ortho-H2 molecules. The comparisons of our MPS calculations with the recent TIQM calculations of Honvault et al.,16,17 which also properly account for the permutation symmetry of the protons, show that in the wide range of collision energies Ec = 10−3 to 0.1 eV our results predict somewhat higher cross and can be typically considered as an upper limit of the otherwise highly oscillating quantum mechanical results. It should be noted that, in the context of the importance of reaction 1 in astrochemical applications,17 the calculated TIQM rate coefficients16,17 with appropriate correction22 are the best available to date. On the other hand, the SQM calculations of Honvault et al.,16,17 as well as SQM and TIQM calculations of GonzalezLezana et al.14 do not fully account for the permutation symmetry of the protons and predict lower cross sections which could be related to the exchange rearrangement mechanism of the protons in reaction 1. We expect that the proposed mean-potential statistical method with proper account of the symmetrization postulate might be very useful in more complicated reactions, in particular for those involving polyatomic molecules, providing, 4577


Mean-Potential Statistical Model and Role of Permutation Symmetry

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