J. Phys. Chem. 1993,97, 8518-8582
8578
ARTICLES Calculated versus Measured Vibrational State Specific Reactivity of H
+ F2
Xavier GimCez, Josep M. Lucas, and Antonio Aguilar Departament de Qulmica Fisica, Universitat de Barcelona, 08028 Barcelona, Spain Antonio Lagand’ Dipartimento di Chimica, Universith di Perugia, Via E k e di Sotto, 8-06100 Perugia, Italy Received: December 30, 1992; In Final Form: May 3, 1993
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Extended approximate three-dimensional quantum calculations of the cross section and product vibrational distributions of the H F2 HF F reaction have been performed in order to carry out a comparison with existing information. It has been found that our calculations agree with quasiclassical and distorted wave results in the range of energy in which they are expected to be more accurate. In addition, both rate coefficients and product vibrational distributions show a satisfactory agreement with the experiment. The followed approach has also allowed a rationalization of important dynamical features of the H F2 reaction.
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TABLE I: State-testate Rate Coefficients Normalized to
I. Introduction
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The H F2(uj) reaction has long been considered a prototype of highly exothermicelementary reactions. This reaction, in fact, in addition to being of practical interest for the modeling of the so-called H F chain reaction,’ has also been a case study for developing the concepts of selectivityand specificityof elementary chemical reaction^.^.' These studies also gave an important contribution to the rationalization of the dynamical behavior of H X2 (with X = halogen) reactions and of the light-heavyheavy (LHH) mass c0mbination.q-7 Available experimental information on the H + F2 reaction is concerned with infrared chemiluminescence emitted by the product H F molecule under different experimental conditions, from which state-specific reactivity (in particular, detailed rate coefficients and product vibrational distributions) can be recovered. Experimental measurements of this highly exothermic reaction were first performed by Jonathan et al.*99 using the “measured relaxation” technique and verified slightly afterward by Polanyi and Sloan4using an “arrested relaxation” method. More recent studies were carried out using fast flow1OJ1 and “chemiluminescence ma~pinglz-1~ techniques. Product vibrational distributions normalized to the maximum derived from these measurements are shown in Table I. They all have a maximum at v’= 6. Differences are mainly due to the estimated efficiency of the relaxation caused by radiation obtainable from Einstein’s coefficients for spontaneous emission. (For the H + F2 reaction accurate values of Einstein coefficients were only made available by the recent chemiluminescence experiments. Recalculation of results corresponding to older experiments using improved new coefficients has been necessary to overcome some of previous experimental errors.)ls17 Several theoretical studies of the H F2 reaction have been reported in the literature. 3D quasiclassical trajectory (QCT) calculations were first performedss9J8-22 to find out whether the proposed potential energy surface (PES)is able to reproduce state-specificexperimental information. QCT results were found to be in good agreement with experimental product vibrational distributionsand toconfirm the already mentioned typical features of LHH systems. Further theoretical work34 focused on the rationalization of the reaction dynamics and isotopic effects
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0022-3654/93/2091-8518$04.00/0
the Maximum u
CMPP
FFb
FP
MRd
ARC
0 1 2 3 4 5 6 7 8 9 10
0.15 0.13 0.27 0.41 0.72 1 0.76 0.46 0.41 0.00
0.07 0.17 0.28 0.59 0.93 1 0.52
0.06 0.12 0.17 0.37 0.76 1 0.62
0.09 0.11 0.13 0.45 0.89 1.oo 0.45 0.20 0.04 0.00
0.12 0.13 0.2s 0.35 0.78 1 0.40 0.26 0.16
.oo
.oo
.oo
.oo
a Chemiluminescence mapping pressured pulse, ref 14. b Fast flow, ref 10. Fast flow,ref 11. Measured relaxation, refs 8 and 9. e Arrested
relaxation, ref 4.
obtained when replacing H by Mu. From a detailed analysis of the main dynamical quantities, the importance of the collinear geometry is confirmed (e.g., the collinear vibrational adiabatic barrier appears to determine the value of the threshold energy). This has motivated extended quantum mechanical collinear Additional dynamical investigations of this reaction have been carried out using quasiclassical27-32 and statistical techniques.33-35 However, up to date, the only existing approximate threedimensional quantum calculation on the H F2 has been performed by Clary and Connor36 using a vibrationally adiabatic distorted wave (VADW) method. Because of the well-known limitations of distorted wave approaches at high energies, we carried out detailed cross section and rate constant calculations using a reactive infinite order sudden (RIOS) technique. Such a choice was motivated by the following facts: (1) vibrations are properly accounted for by the RIOS method, ( 2 ) kinematic characteristics of LHH systems are consistent with RIOS a s s u m p t i o n ~ , 3and ~ ~(3) ~ an extremely efficient vector-parallel version of the RIOS code has been implemented on the IBM 3090/VF 6005 machine at the Computer Center of the University of Barcelona. The paper is organized as follows: The parameters of the calculation are described in section 11. Results and related dynamical implications are discussed in section 111.
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0 1993 American Chemical Society
Vibrational State Specific Reactivity of H
+ F2
The Journal of Physical Chemistry, Vol. 97, No. 33, 1993 8579
11. Parameters of the Calculation The theoretical foundations of the RIOS approach43 adopted here as well as its numerical implementation on vector-parallel computers44 have been already discussed in the literature. Here we sketch only what is necessary to discuss the adopted calculation parameters. The RIOS treatment considers the following reduced dimensionality Schradinger equation:
a2 +-+7+-a2 LA J A
[as:
as;
46-48) was carried out using a pseudoiterativealgorithm.49The resulting unique correspondence between y, and 78 at the matching line allows a droppingof the arrangement channel label A. Finally, from the state (v) to state (u') single 1 fixed-angle S matrix elements Sl,(y) calculated at the asymptote by imposing scattering boundary conditions on the propagated solution, the integral reactive u,/(E)
(as usual, k,Z = 2 4 E - e,)/h2) as well as the differential
SA :s
where p is the reduced mass of the system, SAand SAare a suitable pair of propagation and bound orthogonal coordinates (e.g., atomdiatom and diatom mass scaled Jacobi coordinates) for channel A (A = a for the reactant channel and h = B or 8' for the product channels),E is the total energy, and YAis the fixed collision angle formed by SAand SA. Since all kinds of angular motions are decoupled in a RIOS approach, LA = l ~ ( l ~1)+and JA= j A ( j x =+1) are mere atom-diatom orbiting and diatom rotating constant quantum labels chosen to bej , - 1, = 0 and incrementing I, = j p in steps of one till convergence is reached. Following Light and Walker,45 the integration of eq 1 is simplified by segmenting the propagation coordinate into 100 sectors of width 6 = 0.0305 a. in the reactant channel and 200 sectors of width 6 = 0.0225 a0 in the product channel. The largest number of sectors adopted in the product channel is motivated by the fact that for the products the potential energy variation is more dramatic than for reactants because of the strong exoergicity of the reactive process (4.48 eV). Sector equations
+(SAPAFYJ
0 (2)
(i is the sector number and V(S$',,Y,) is the value of the potential at the midpoint of the ith sector that is assumed to be constant along the propagation coordinate) are solved by expanding locally the global wave function +(SX,SA;YA) in terms of products of a propagation x b ( S ~ ; y and ~ ) a bound I& (s,;SA,YJ function. The bound &(sASA,TJ component of the global wave function and related eigenvalues are determined by solving numerically the one-dimensional eigenvalue problem
over a grid of 1000 points in SA. In eq 3 Si is the value of the propagation coordinate at the midpoint of sector i. Then, by substituting the expansion intoeq 1, the set of coupleddifferential equations
is obtained. Equations 4 are integrated using an intrasector diabatiointersector adiabatic44.45 procedure from the line dividing the reactant and product channels to the asymptotes. The outer integration limits have been set as Sa = s8 = 40 A. In eqs 4 D~(E;YA) is the fixed collision angle, total energy, and orbital quantum number coupling matrix whose straightforward dependence from E and l allows its simple determination for parallel runs44 on 1 and E. The matching of the reactant and product solutions at the dividing line (chosen to satisfy the criteria of refs
cross sections can be evaluated. From the cross sections, the reactant vibration-rotation state to all product states rate coefficients, defined as
where a,@) = u,d(E) and ke is the Boltzmann constant, can be also derived. Due to the difficulty of calculating an accurate global ab initio PES for this system, only semiempiricalpotential energy surfaces were used for dynamical calculations. Incomplete (i.e., insufficient to the end of fully spanning the configuration space) CI calculationswere carried out by Schaefer and co-workers.sOThese calculations did not reproduce the experimental exothermicity and the equilibrium distances. Essentially similar problems affected also POL-CI cal~ulations.5~Other analysis of the electronic structure of the H + F2 system and of its reactive properties were carriedout using LEPS,599J99DIM,5253 and DIM3Cs4potential energy surfaces (includingin some cases electronic excited states and spin-orbit coupling).52,53 Most of previous dynamical calculations were carried out on the LEPS surface of Jonathan, Okuda, and Timlin (JOT)? Because of this and the ability of such a PES to lead to agreement between calculated and experimental properties of the H F2 reaction, we used the JOT surface for our RIOS calculations. The main difficulty in carrying out quantum calculations for the H F2 reaction is represented by the narrow spacing of the F2 levels and the sudden change in the nature of the potential energy surface at the transition state associated with a jump of 4.5 eV in a small range of internuclear distances. This implies that a high number of vibrational basis functions need to be included in the expansion of the global wave function even at moderately high collision energies. This not only makes it extremely difficult to extend the calculations to high energies but also asks for a careful check of the convergence with respect to the basis set size. In the investigated energy range (the total energy E was increased in steps of 0.01 eV up to 4.9 eV above the asymptotic minimum of the product channel) convergence was reached using a basis set of dimension 30. The suddenvariation of the reaction channel past the transition state required also, as already mentioned, a doubling of the used sectors in the product region when compared with the reactant one. The LHH mass combination of H F2 allowed an efficient use of circular coordinatesto propagate the solution in the strong interaction region. In fact, the generalized skewing angle introduced by the RIOS treatment not only is quitelarge (80.87')
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Gimdnez et al.
8580 The Journal of Physical Chemistry, Vol. 97, No. 33, 1993 20
1.0
0
8 > 0.5
0.0
3
6
12
9
V’
Figure 2. RIOS (squares), QCTs (open circles), VADW36(triangles),
and e~perimental~~ (solid circles) vibrational distributions normalized to the maximum, for F2 in the ground vibrational state. Ew = 0.105 eV.
2o 0.0
0.1
0.3
0.2
0.4
E,, I eV Figure 1. RIOS (squares), QCT (Circles),and VADW (triangles) cross sections, for F2 in the ground vibrational state, plotted as a function of the translational energy Ew. The energy zero was set at the reactant asymptotic minimum. A logarithmic and a linear scale for the y axis were used in the lower and upper panel, respectively.
TABLE Ik Measured and Calculated Rate Coefficients (in 10-12 cm3 molecule-1 s-l) for the H F2 Reaction T,K exp ex9 QCTf VTSTd RIOS
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300 353 400
3.6 6.5 9.8
1.7 2.9 4.2
2.5
4.5 8.6 14.0
6.5 16.5 34.4
Experiment,ref 57. b Experiment,ref 58. Quasiclassicaltrajectories, ref 30. Variational transition state, ref 35. e Present work. a
f
for the collinear configuration but also gradually increases when the collision angle decreases. 111. Analysis of the Results
The first step of our analysis is a comparison of the RIOS results with those obtained from other approximate 3D calculations. As already mentioned, results of QCT’O and VADW36 calculations carried out on the same LEPS surface are available from the literature. For comparison, the v = 0 RIOS, QCT, and VADW cross section values are plotted in Figure 1 (squares, circles,and triangles,respectively)as a function of the translational energy Etr. As clearly shown by the figure, the RIOS and the QCTvalues agree reasonably well while VADW ones show larger deviations. To better illustrate relative deviations in both largeand small-amplitude regions, we used a linear vertical scale in the upper panel and a logarithmic one in the lower panel. The figure shows that RIOS results agree with distorted wave ones at very low energy (where the VADW technique is believed to be accurate) and with quasiclassicalones at high energy (where the QCT technique is less affected by classical mechanics limitations) indicating that, on the whole, they give an overall more accurate picture of the reactivity of H + Fz. A comparison of calculations with the experiment is possible by computing reactive rate coefficients. Calculated values are given in Table I1 where for comparative purposes also experimenta1,57.58VTST,35 and QCT30 data are reported. Both experimentaland VTST correspond to thermalized rate constants, Le., rotationally and vibrationally averaged, whereas QCT and RIOS results are state specific, for the ground vibrational and rotational levels. Despite this, the general agreement between theory and experiment can be considered satisfactory.
”
0.0
0.1
0.2
0.3
0.4
EIeV Figure 3. RIOS v = 0 (squares), v = 1 (circles), and v = 2 (triangles)
cross sections plotted as a function of the energy E. Energy zero was set at the reactant asymptotic minimum. A more detailed comparison of RIOS calculations with the experiment can be carried out by plotting vibrational distributions of products (normalized to the maximum). As shown by Figure 2, where the normalized product vibrational distributions for the H Fz(v-0) reaction have been plotted as a function of the product vibrational number, at a translational energy Eu = 0.105 eV, though largely different in absolute value, the three distributions have a quite similar shape. In particular, they have a common maximum at v’ = 6. The RIOS values quite closely agree with VADW results for v’>6 (right-hand side of the figure) while they better agree with QCT results for J
