Quasiclassical trajectory study of the F + H2 system. Rate constants, kinetic isotope effects and energy partitioning among reaction products

Chemical Physics ELSEVIER

Chemical Physics 195 (1995) 291-303

Quasiclassical trajectory study of the F + H 2 system. Rate constants, kinetic isotope effects and energy partitioning among reaction products Efrat Rosenman, Avigdor Persky Department of Chemistry, Bar-llan University, 52900 Ramat Gan, Israel

Received 30 November 1994

Abstract Quasiclassical trajectory calculations were carried out for the reactions F + H e, F + D 2, and F + HD, using two potential energy surfaces T5A and 6SEC. The results which include rate constants and kinetic isotope effects as a function of temperature, isotopic branching ratios for F + HD as a function of collision energy, and the energy partitioning and vibrational state distributions of the products at room temperature, are compared with experimental data. For most of the quantities under study, the results for the 6SEC surface are in qualitative agreement with experiment, as opposed to the results for the T5A surface. The conclusions from the present study concerning the quality of the 6SEC surface are consistent with the conclusions of Aoiz et al. which are based mainly on calculations of vibrationally state resolved differential cross sections and of vibrational distributions of products, for specific collision energies. 1. Introduction The reaction F + H 2 is one of the most extensively studied reactions in recent decades, both experimentally and theoretically. Early experiments concentrated mainly on the determination of rate constants [1,2], kinetic isotope effects [3-5] and the energy partitioning among reaction products under thermal conditions [6-8] (for a review of early measurements and calculations see Ref. [9]). More recent experiments include the determination of vibrationally state resolved integral and differential cross sections by molecular beams [10-12], the study of the properties of the F - H - H transition state by measurements of the photodetachment spectra of the ion FH~- [13-16], and the determination of the isotopic branching ratio for F + HD by a two-laser experiment [17]. One of the most interesting observa-

tions in the molecular beam experiments [10,11] was the appearance of a sharp forward peak in the differential cross sections for the products in one of the vibrational states, while the scattering was predominantly backwards for all the other states. This state was H F (v' = 3) for F + H 2 ~ H F + H and F + HD ~HF+D and D F ( d = 4 ) for F + D 2 ~ D F + D and F + DH ~ D F + H. These experiments also showed vibrational population inversion with the most populated states being H F ( v ' = 2) and D F ( v ' = 3). The forward scattering was interpreted in terms of quantum mechanical resonances. Concurrently with the experimental studies of the F + H 2 system, extensive theoretical studies were also conducted. The most critical element in such studies is the availability of a reliable potential energy surface, which is essential for the calculations. Over the years more and more accurate potential

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292

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

energy surfaces have been developed (for a summary see, for example, Ref. [18]). A surface which has been employed extensively is the semiempirical LEPS surface M5 developed by Muckerman [19]. Calculations on this surface showed only partial success in reproducing the experimental data. Quasiclassical trajectory calculations (QCT) [19] of the energy partitioning among reaction products and of vibrational state distributions gave results in qualitative agreement with experiment. However, QCT calculations of differential cross sections on this surface [20,21] did not show any significant forward scattering for any vibrational state of the products, in contrast to the experimental findings. Quantum mechanical calculations (QM) [22-24] showed resonance effects for v ' = 2, rather than for c ' = 3 as found experimentally. This behavior was attributed to the existence of a high adiabatic barrier for the formation of products in the v' = 3 state. An improved potential energy surface T5A was developed by Truhlar and co-workers [25,26]. This surface does not have an adiabatic barrier for the formation of HF (v' = 3) and DF (v' = 4). Neuhauser et al. [27] used this surface to calculate QCT and QM integral cross reactions for F + H e, taking into account the initial rotational distribution of the H 2. The QCT results were found to be in good agreement with experiment. Less satisfactory agreement was obtained for the QM results. Vibrationally state resolved differential cross sections were calculated by Hayes and Walker [28,29] employing the approximate bending corrected rotating linear model (BCRLM). Good agreement with experiment was obtained for F + D 2 ~ DF + D and F + DH ~ DF + H, partial agreement was obtained for F + H 2 HF + H, but disagreement was found for F + HD HF + D. In another study [30,31] differential cross sections were calculated for F + H 2 in hyperspherical coordinates and qualitative agreement with experiment was obtained. However, vibrational branching ratios of the products showed preference to the formation of HF (v' = 3) in contrast to the experimental findings. Takayanagi and Sato [32] developed a modified LEPS potential energy surface TS which has a bent transition state and a flat bending potential. Their QCT calculations on this surface for F + H 2 (v = 0, j = 0) reproduced qualitatively the forward scattering for HF (v' = 3) and the vibrational state distribution

of the products and thus proved that the forward scattering found experimentally for HF O, ' = 3) is not necessarily due to quantum mechanical resonances, but may be associated with the features of the potential energy surface. QM calculations of the photoelectron spectra of the ion FH 2 for this surface [16] gave results in reasonable agreement with experiment. QCT calculations on this surface by Aoiz et al. [33] for F + H D ( v = 0 , j = 0 ) at a collision energy of 1.98 k c a l / m o l showed that the most populated vibrational states of the products are HF (v' = 2) and DF (v' = 3), in qualitative agreement with experiment. They also showed forward scattering for HF ( v ' = 3). However, the calculated rotational excitation of the products and the absolute values of the integral cross sections are probably too high. Truhlar and co-workers have recently developed two new potential energy surfaces 5SEC [34] and 6SEC [35]. Each of these surfaces has a bent transition state and a flat bending potential. QCT calculations of differential cross sections using the 5SEC surface by Aoiz et al. for F + H 2 and F + D 2 [36] and for F + HD [33] showed forward scattering for HF (v' = 3) and DF (v' = 4). However, the branching ratio O'R(V'= 3)/O'R(V' = 2) for F + H 2 was much higher than the experimental ratio, showing preferred formation of c ' = 3, in contrast to the experimental results. QM calculations on this surface [34] gave also a high value for this ratio. The photoelectron spectra simulated using this surface [16] were in less satisfactory agreement with experiment than the spectra calculated for the TS surface. The 6SEC surface is similar to the 5SEC surface, but has a high adiabatic barrier for the formation of HF ( v ' = 3). QM calculations on this surface for F + H 2 (v = 0, j = 0) [35] showed that, except for the lowest experimental collision energy, the formation of HF (c,' = 2) is favored over the formation of HF ( v ' = 3), in better agreement with experiment than the results for the 5SEC surface. QCT calculations on this surface by Aoiz et al. for F + H e (v = 0, j = 0) [37] also showed preference to the formation of HF (v' = 2). The calculated differential cross sections showed forward scattering for HF (v' = 3), in qualitative agreement with experiment. In a very recent paper [38] Aoiz et al. report results of QCT calculations for F + H 2 (v = 0, j = 0 - 2 ) for the TS and 6SEC surfaces. The results for the specific initial rotational states j = 0 - 2 were

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

used to calculate differential cross sections and vibrational branching ratios of products for F + n-H2, giving proper weights to the various initial j states, according to the distribution in the experiments of Neumark et al. [10]. They also calculated rate constants at 298 K. From the comparison of the calculations with the experiments they concluded that both surfaces perform well for the calculation of vibrationally state resolved differential cross reactions, although the forward peak for u ' = 3 seems to be more pronounced in the calculations than in the experiments, especially for the 6SEC surface. Both surfaces also yield reasonable agreement with the experimental vibrational branching ratios of products, though O'R(d = 3)/erR(t/ = 2) is overestimated and CrR(V'= 1)/O'R(1"=2) is underestimated as compared with the experiment. They also point out the reasonable agreement which was obtained between QM calculations for the TS surface and the experimental photoelectron spectra for FH 2 [16]. No calculations of the photoelectron spectra have been carried out for the 6SEC surface. However, Aoiz et al. assume that such calculations would lead to results similar to those obtained for the 5SEC surface, which are in less satisfactory agreement with experiment than the results for the TS surface. This assumption is based on the fact that the transition state region, which is the region probed by the photoelectron spectra, is the same for the 5SEC and 6SEC surfaces. Aoiz et al. conclude that the TS surface performs somewhat better than the 6SEC surface, when judged by the results for differential cross sections, vibrational distribution of products and the photoelectron spectra of FH 2 . However, the rate constant at 298 K calculated for the TS surface is in serious disagreement with experiment (several times larger), and therefore this surface is inadequate for the description of the F + H 2 reaction. The 6SEC surface, on the other hand, predicts a rate constant in reasonable agreement with experiment. The large rate constant for the TS surface is attributed to its very low barrier ((I.18 kcal/mol). Very recently a new ab initio potential energy surface SW was developed by Stark and Werner (not yet published [39]). QM calculations of the photoelectron spectra for FH 2 using this surface [40] gave results in very good agreement with experiment. QCT calculations for F + H 2 ( v = 0, j = 0-2) on

293

this surface were carried out by Aoiz et al. [18]. The calculations showed that vibrationally state resolved integral and differential cross sections, and vibrational branching ratios, are in good agreement with experiment, though the ratios O'R(t/ = 3)/o-R(d = 2) are systematically too low and the ratios OR(t,' = I)/O'R(V' = 2) are too high (just the opposite to the results for the 6SEC surface [36]). The main difference from the experimental results, was the magnitude of the forward peak which was found to be lower and confined to a narrower range of scattering angles for the QCT calculations than found experimentally. In addition, in contrast to the experimental results and the QCT results for the 6SEC surface, which indicated a decrease in the forward peak with the initial rotational state, the calculations for the SW surface showed an increase. Aoiz et al. conclude that it is likely that the SW surface is more accurate than all the previous surfaces. The experimental data against which predictions based on the potential energy surfaces mentioned above have been tested until now, consisted mainly of integral and differential cross sections and of vibrational branching ratios, for some specific collision energies, and of the photoelectron spectra of FH e . The comparison with kinetic data was limited to the rate constant at 298 K. In order to learn more about the quality of these surfaces it is desirable to carry out such tests also with regard to other available experimental data, and this is the goal of the present study. Here we present results of QCT calculations for the potential energy surfaces T5A and 6SEC. The results which are compared with experimental data include rate constants and kinetic isotope effects as a function of temperature, the energy partitioning and vibrational branching ratios of products under thermal conditions at 300 K and isotopic branching ratios O-R(F+ HD)/o-R(F + DH), for specific collision energies and a thermal distribution of initial rotational states of HD at 300 K.

2. Results and discussion 2.1. F + H 2 and F + D 2 : cross sections, rate constants and kinetic isotope effects

The computational procedure was similar to the procedure used by us in earlier QCT studies of

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

294

atom-molecule reactions (for example, Ref. [41], and references therein). Cross sections for the reactions F + H 2 ( v = 0 , j = 0 - 4 ) and F + D 2 ( u = 0 , j = 0 - 4 ) were calculated for the potential energy

surfaces T5A and 6SEC. For each set of initial conditions (collision energy and rotational state) usually 8000 to 10000 trajectories were calculated. Based on preliminary calculations, the maximum

i

(a) T5A: F+H 2 (v=0,j)

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Err (kcal/mol) Fig. 1. Reaction cross sections as a function of collision energy for the reactions F + H 2 (v = 0, j = 0 - 4 ) and F + D 2 (v = 0, j = 0 - 4 ) for the potential energy surfaces T5A and 6SEC. (a) T5A, F + H 2 (b) T5A, F + D 2 (c) 6SEC, F + H 2 (d) 6SEC, F + D 2. ( D ) j = 0; ( 0 ) j=l;(zx)j=2;(ll) j=3;(v) j=4.

295

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

impact parameter, in each case, was chosen to be somewhat larger than the largest one leading to reaction. The initial distance between the F atom and the center of mass of the diatomic molecule was 6 ,~ in all the calculations. Calculated excitation functions (reaction cross sections as a function of collision energy) for F + H 2 (v=0, j=0-4) and for F + D 2 ( t o = 0 , j = 0 - 4 ) , for the potential energy surfaces T 5 A and 6SEC, are presented in Fig. 1. As can be seen from Fig. 1, the influence of rotation on the cross sections is distinctly different for the two surfaces. For the T5A surface rotation tends to hinder reaction, except for an increase between j = 0 and j = 1, whereas for the 6SEC surface rotation is beneficial to reaction, especially between j = 0 and j = 2, with no significant influence for higher values of j. A behavior similar to that found for the T 5 A surface, but with no increase between j = 0 to j = 1, was obtained for the M5 surface (Ref. [19], Fig. 2). Excitation functions for F + H 2 (v = 0, j = 0 - 2 ) for the range of collision energies between threshold and approximately 7 k c a l / m o l , for the 6SEC and TS potential energy surfaces, were calculated by Aoiz et

i

i

i

i

al. [38]. They showed that in contrast to the richness of features exhibited for the TS surface, the excitation functions for the 6SEC surface are relatively featureless, increasing monotonically with collision energy (Ref. [38], Fig. 2). The excitation functions calculated by us (Fig. 1) are, in general, in good agreement with those of Ref. [38]. However, in our study many more trajectories were calculated and the range of collision energies was wider than in Ref. [38] and this allows the observation of features in some of the excitation functions which were not observed in Ref. [38]. Thus, for F + H 2 (v = 0, j = 0), o-R increases monotonically up to a collision energy of about 4 k c a l / m o l , remains unchanged, or even decreases slightly, between 4 and 5 k c a l / m o l , increases again up to 6.5 k c a l / m o l and then levels off, or increases slightly, for higher collision energies. Similar behavior is observed also for F + D 2 ( v = 0 , j = 0 ) and to a lesser extent for F + D e (v=0, j=2). Vibrationally state resolved excitation functions for F + H 2 (v = 0, j = 0) --+ H F (v' = 1 - 3 ) + H and for F + D 2 (to = 0, j = 0 ) --+ D F (v' = 1 - 4 ) + D for the 6SEC surface are shown in Fig. 2. It can be seen

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6

8

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Etr(kcal/mol) Fig. 2. Vibrationally state resolved cross sections as a function of collision energy for the reactions F + H 2 (v = 0, j = 0 ) ~ HF (v' = 1-3) + H (left panel) and F + D2 (v = 0, j = 0) ~ DF (v' = 1-4) + D (right panel) for the 6SEC potential energy surface.

296

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

that the features observed in Fig. 1 for total cross sections become more pronounced in the vibrationally state resolved excitation functions for HF ( t / = 2) and DF ( v ' = 3). They also appear for DF ( v ' = 2), but less prominently than for DF ( J = 3). These are real effects, beyond the statistical errors of the calculations, and this is confirmed by the qualitative similarity between the features in the excitation functions for F + H 2 and F + D 2 in Figs. 1 and 2. In contrast to this behavior, Aoiz et al. report relatively featureless excitation functions for all 5,' states (Ref. [38], Fig. 2). However, a careful examination of their results reveals similar trends to those observed by us for all v' states, although much less pronounced. Our vibrationally state resolved cross sections for F + H 2 (~, = 0, j = 0) ~ HF (~,' = 1 - 3 ) + H for the 6SEC surface are qualitatively similar to the results of Aoiz et al. for the TS surface (Ref. [38], Fig. 2), although the features are more pronounced and are shifted to lower collision energies for the TS surface. Thus, the maximum and minimum for v' = 2 for the TS surface occur at collision energies around 1.3 and 2.3 kcal/mol, respectively, as compared to 3.5 and 5

kcal/mol, respectively, for the 6SEC surface, and the maximum for c ' = 3 for the TS surface occurs around 1.3 kcal/mol, as compared to 4 k c a l / m o l for the 6SEC surface. The shift to lower collision energies for the TS surface is probably associated with the lower barrier for this surface. For L/= 1, both surfaces predict a monotonic increase with collision energy. The beneficial role of rotation was demonstrated experimentally [10] by measuring a larger cross section for F + n-H e than for F + p - H 2 (larger by a factor of 1.23 at a collision energy of 1.84 kcal/mol). QCT calculations by Aoiz et al. [18] on the new SW surface [39] for the same collision energy, showed an increase by a factor of 1.16, in good agreement with experiment. To simulate the experimental conditions they used weights of 0.80 and 0.20 for j = 0 and j = 2, respectively, for P-H2, and weights of 0.25 and 0.75 for j = 0 and j = 1, respectively, for n-H2. Using the same weights, we calculated an increase by a factor of 1.18 for the T5A surface and by a factor of 1.29 for the 6SEC surface, also in good agreement with experiment. The increase in the case

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1000/'T (K-~) Fig. 3. Semi-logarithmic plots of the rate constants kF+ n2 (left panel) and kF+D2 (right panel) as a function of 103/T. Experimental results: ([]) Ref. [43]; (C)) Ref. [1]; (,',) Ref. [2]. Results of calculations: (Q) QCT-T5A, this work; (11) QCT-6SEC, this work; ( • ) QCT-M5, Ref. [19], (*) VTST-T5A, Ref. [26].

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

of the T5A surface is due to the increase in cross sections between j = 0 to 1, before the decline for higher values of j. The cross sections presented in Fig. 1 were used to calculate rate constants, taking into account the thermal distributions of collision energies and of initial rotational states. The multiple surface coefficient f=[exp(-A/kT)+2] -1, where A is the F(2p3/2,1/2) spin-orbit splitting of the F atom [42], was included in the calculations. The QCT rate constants for the surfaces T5A and 6SEC calculated in the present study, and those calculated by Muckerman [19] for the M5 surface, are compared with experimental results of three groups in Table 1 and in Fig. 3 (these are the three most recent determinations of rate constants as a function of temperature for this reaction). The experimental results in Fig. 3 are the actual results reported in the literature, while those in Table 1 were calculated from the Arrhenius fit to the experimental rate constants. This was done in order to compare absolute values of calculated rate constants with measured rate constants of various groups, for the same set of temperatures. Table 1 and Fig. 3 include also results of calculations by the variational transition state theory method (VTST) [26] for the T5A surface. Arrhenius parameters for each set of kinetic data are also given in the table. Examining the results for F + H 2 (Table 1 (upper part) and Fig. 3a) one can see that all the QCT rate constants are lower than the experimental rate constants, being lowest for the M5 surface and highest for the 6SEC surface. The rate constants for the 6SEC surface approach the experimental results at high temperatures, but the discrepancy becomes larger as the temperature is lowered. The activation energy for this surface is higher than the average of the three experiments by about 0.25 kcal/mol. A QCT rate constant for the 6SEC surface at 298 K has been calculated by Aoiz et al. [38]. The value obtained from their calculations, after including the multiple surface factor, is 1.73 × 10 -11 cm 3 molecule 1 s 1, in very good agreement with the value calculated by us. The QCT rate constants for F + D 2 (Table 1 (lower part) and Fig. 3b) show the same trend as found for F + H 2, being lowest for the M5 surface and highest for the 6SEC surface. For this surface

297

Table 1 Theoretical and experimental rate constants

for the reactions

F + H 2 and F + D 2 a T (K)

Experiment b

VTST

WH c SBA d HBGM e T5A f

QCT T 5 A g M5 h 6SEC g

F+H 2 ~ HF+H 200 1.14 -

-

0.87

0.53

0.21

0.6l

250

1.76

1.83

-

1.38

0.83

0.40

1.13

298

2.33

2.48

2.91

1.85

1.12

0.60

1.69

350

2.89

3.14

3.92

2.33

1.42

0.83

2.32

400

-

-

4.85

2.75

1.70

1.04

2.92

450

-

-

5.73

3.13

1.97

1.25

3.50

500

-

-

6.54

3.47

2.22

1.44

4.06

A a

8.73

5.66

5.15

1.18

0.91

0.95

1.26

1.25

F + D 2 --~ D F + D 200 0.57 -

-

0.45

0.27

0.10

0.36

250

0.99

0.61

-

0.73

0.45

0.20

0.70

298

1.41

0.95

1.39

1.00

0.63

0.31

1.08

350

1.86

1.33

1.88

1.28

0.83

0.44

1.51

400

-

-

2.34

1.52

1.02

0.56

1.92

450

-

-

2.76

1.74

1.21

0.67

2.33

500

-

-

3.16

1.94

1.40

0.78

2.71

A a Ea i

9.1 1.10

9.3 1.35

10.6 1.20

5.14 0.96

4.07 1.09

3.01 1.34

10.3 1.33

Ea i

a

10.0 0.86

12.0 0.93

21.6

14.1

Units of 10-11 cm 3 m o l e c u l e - 1 s 1.

b Values were calculated from the reported Arrhenius fits to the experimental data. c R e f [1]. d Ref. [43].

c Ref. [2].

f Values calculated from an Arrhenius fit to variational transition state theory results reported by Steckler et al. [26] for the temperature range 2 0 0 - 5 0 0 K. g This work. h Values calculated f r o m an Arrhenius fit to results reported by M u c k e r m a n [19] for the temperature range 2 5 0 - 4 5 0 K. i Units of k c a l / m o l .

(6SEC) the calculated results are in generally good agreement with the experimental results of Stevens et al. [43] at all temperature and with the results of the two other groups [1,2] at high temperatures, becoming lower than their results at low temperatures. The calculated activation energy is lower than the average experimental activation energy only by about 0.10 kcal/mol, which is less than the differences between the various experiments. As can be concluded from Table 1 and Fig. 3, comparing the QCT results for the potential energy surfaces M5, T5A and 6SEC, the results for the 6SEC surface are the closest to the experimental data. If tunneling is important for the F + H 2 reac-

298

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

height of the barrier increases very moderately with the change in the F - H - H angle. Because of these differences, reactive collisions will occur for wider ranges of impact parameters and of F - H - H angles on the 6SEC surface than on the other surfaces, leading to higher cross sections and higher rate constants and also to more forward scattering. The data in Table 1 and Fig. 3, provide an opportunity to compare theoretical kinetic results obtained by two different methods (VTST and QCT) for the same potential energy surface (T5A). It can be seen that the VTST rate constants are much higher than the QCT rate constants for both isotopic reactions F + H 2 and F + D 2. One might suppose that these differences can be attributed, to a large extent, to quantum effects, such as tunneling, which are included in the VTST calculations, but not in the QCT calculations. However, if this were the case, we would expect the ratio k(VTST)/k(QCT)to be much larger for F + H 2 than for F + D 2, and that this ratio would decrease significantly with temperature. The results presented in Table 1 show that the ratio for the two isotopic reactions is similar (1.65 for F + H 2

tion on this surface, one may expect that including tunneling corrections will improve the agreement with experiment. Such corrections will increase the rate constants (especially for low temperatures) and will lower the activation energy for both reactions F + H 2 and F + D 2. The effect will be more significant for F + H 2 than for F + D 2, as needed in order to improve the general agreement with experiment. The higher reactivity found for the 6SEC surface than for the M5 and T5A surfaces, as indicated by the higher values of rate constants (Table 1 and Fig. 3) and of cross sections (Fig. 1 of this paper and Fig. 2 of Ref. [19]), can probably be attributed to significant differences in the properties of these surfaces (for a discussion of the properties of the surfaces see, for example Refs. [18,26,35]. The M5 and T5A surfaces have linear transition states (barrier heights 1.06 and 0.94 kcal/mol, respectively) with relatively tight bending potentials, which means that the barrier height increases rapidly with the F - H - H angle. On the other hand, the 6SEC surface has a bent transition state (104 °, barrier height 0.97 kcal/mol) with a very flat bending potential, which means that the

4.0

i

i

i

i

(b) F+HD/F+DH

(a) F+H21F+D 2

3.0

El

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g

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EXP T5A- QCT M5- QCT 6SEC- QCT T5A- VTST

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0 ffl

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0

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1.0 0.9 2

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I 4

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6

1000/T (K-I) Fig. 4. Semi-logarithmic plots of the kinetic isotope effects kF+ bi2//kF+ DE (left panel) and kF+ HD//kF+ DH (right panel) as a function of 103/T. Experimental results: (C3) Ref. [3] and Ref. [4] for kF+ H2/kF+D2 and kF+ HD/kE+ OH, respectively. Results of calculations: ( Q ) QCT-T5A, this work; ( • ) QCT-6SEC, this work; ( • ) QCT-M5, Ref. [19], ( * ) VTST-T5A, Ref. [26].

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303 Table 2 Theoretical and experimental values of the kinetic isotope effect

kF+ H2// kF+ D2 T (K)

163 200 239 2t~8 417

AH/A ° e AE a f

Experiment a

VTST

QCT

Y5Ab

T5A c

M5 d

6SEC c

3.27 2.52 2.26 1.91 1.69

1.98 1.92 1.89 1.85 1.79

2.16 1.99 1.88 1.77 1.65

2.14 2.05 1.99 1.93 1.87

1.73 1.65 1.60 1.55 1.50

1.04 -0.37

1.70 --0.05

1.39 --0.14

1.71 --0.07

1.37 -0.08

a Persky [3]. b Results of variational transition state theory reported by Steckler el al. [26]. c This work. d Values calculated from an Arrhenius fit to the rate constants reported by Muckerman [19]. e Ratio of pre-exponential factors for F + H 2 and F + D 2. f Activation energy difference E~(F + H 2) - Ea(F + D 2) in kcal/mol.

and 1.59 for F + D 2 at 298 K). They also show only a slight decrease in this ratio with temperature for F + H 2 (from 1.64 at 200 K to 1.56 at 500 K) and a larger decrease for F + D 2 (from 1.67 at 200 K to 1.39 at 500 K). These observations indicate that the differences between the results for the VTST and QCT calculations are not due to tunneling, but rather to differences in the fundamental assumptions on which these methods are based (for reviews of the VTST and QCT methods see Refs. [44] and [45], respectively). Values of the kinetic isotope effect kF+ H 2 / k F + D 2 as a function of temperature, calculated in the present study for the T5A and 6SEC surfaces, as well as QCT results for the M5 surface [19] and VTST results for the T5A surface [26] are compared with experimental data [3] in Table 2 and in Fig. 4a. The experimental results of Ref. [3] were obtained from a direct determination of the kinetic isotope effect at the temperatures listed in the table. A later determination of the kinetic isotope effect by Grant and Root [5] using a different method gave results in very good agreement with these results. As can be seen from Table 2 and from Fig. 4a, the calculated isotope effects, except for some results at the high temperature range, do not agree satisfactorily with the experimental data. The lowest values

299

were obtained for the 6SEC surface. All the slopes of the calculated results (indicating activation energy differences Ea(F + H 2) - Ea(F + D2)) are much smaller than the experimental slopes. If tunneling is important for any of these surfaces, then it may be expected that adding tunneling corrections to the QCT results would increase the slopes and absolute values of the isotope effect and thus reduce the discrepancy between the calculations and experiment. The VTST results already include tunneling corrections and their agreement with experiment is not better than that of the QCT results.

2.2. F + HD: Isotopic branching ratios and intramolecular kinetic isotope effects Cross reactions for the two channels of the reaction of F atoms with HD, F + HD ~ HF + D and F + DH ~ DF + H, were calculated for the T5A and 6SEC surfaces for specific collision energies and thermal distributions of rotational states of HD ( j = 0-5). For each temperature and collision energy usually 25 000 to 30 000 trajectories were calculated. Isotopic branching ratios OrR(F + H D ) / o - R ( F + DH), as a function of collision energy at 300 K, are presented in Fig. 5, where they are compared with the experimental results of Johnston et al. [17]. As can be seen from Fig. 5, the calculated branching ratios for both surfaces are lower than the experimental ratios. However, the dependence on collision energy is qualitatively similar. Both experiment and calculations show a decrease with collision energy up to about 6 kcal/mol, with no significant change above this energy. At collision energies above approximately 4 kcal/mol, the two surfaces predict nearly the same branching ratios, showing preference to the formation of DF, in contrast to the experimental findings. (Approximate values for the M5 surface evaluated from Fig. 6 of Ref. [19] show even smaller branching ratios.) At lower energies, which contribute more to thermal rate constants (for the 6SEC surface, collision energies up to 4 kcal/mol contribute approximately 95% to the values of kF+HD and kF+ou at room temperature), the branching ratios for the 6SEC surface increase more rapidly with decrease in energy than for the T5A surface, predicting preferential formation of HF below a collision energy of about 3 kcal/mol. It seems that the

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

300

2.5

Table 3 Rate constants for the two channels of the F + H D reaction ~

i

F+HD/F+DH i

EXP T5A 6SEC

20

O

rr" 1.5 c tO ¢-

[]

[]

1.0

0.5

oo

I

2

[31

•

•

I

6SEC h

F+HD

F+DH

F+HD

F+DH

F+HD

F+DH

159 168 222 298 413

0.96 1.13 2.27 3.95 6.20

0.92 1.08 2.17 3.76 5.89

0.50 0.60 1.35 2.54 4.23

0.46 0.56 1.32 2.61 4.50

1.17 1.43 3.48 7.20 13.2

0.86 1.06 2.66 5.66 10.6

I

OOoo%%a 4

T5A b

a Units of 10 -12 cm 3 molecule i s 1 b This work. c Values calculated from an Arrhenius fit to results reported by Muckcrman [19] for the temperature range 1 5 0 - 4 5 0 K.

• []

rn

M5 "

T (K)

I

6

I

8

110

112

Etr (kcal/rnol) Fig. 5. Isotopic branching ratio C r R ( F + H D ) / t r R ( F + D H ) as a function of collision energy, for a thermal distribution of rotational states of HD at 300 K. ( • ) experimental results, Ref. [17]; ( O ) Q C T - T 5 A , this work; ( D ) Q C T - 6 S E C , this work. The rectangles surrounding the experimental points indicate the uncertainties in the collision energy and the error limits of the branching ratios.

calculated ratios for the 6SEC surface approach the measured ratios at low collision energies and this is confirmed by the fact that the branching ratio 1.52 _+ 0.23 measured by the same research group [46] for a thermal distribution of collision energies at 300 K (the intramolecular kinetic isotope effect), agrees quite well with the value 1.27 calculated for the 6SEC surface (Table 4). The calculated cross sections for the two branches of the F + HD reaction, already properly weighted over initial thermal distributions of rotational states, were used to calculate values of the rate constants kF+ HD and kF+DH, and of the intramolecular kinetic isotope effect kv+ m~/kv+Du. Values of the individual rate constants kv+ HD and kv+ HDare presented in Table 3. Values of the isotope effect kF+HD/kF+DH are presented in Table 4 and in Fig. 4, where they are compared with QCT results for the M5 surface [19], VTST results for the T5A surface [26] and with experimental results [4]. In addition to the experiments of Ref. [4], which are included in Table 4 and

in Fig. 4, two other measurements of this kinetic isotope effect were carried out at room temperature [6,46], giving results in good agreement with those of Ref. [4]. It can be seen that out of the QCT results presented in Table 4 and in Fig. 4, the best agreement with experiment was obtained for the 6SEC surface. The values calculated for this surface are lower than the experimental values only by 10 to 12 percent in the temperature range of the experiments. The results for the two other surfaces are in disagreement with experiment. For the T5A surface, the kinetic isotope effect is close to unity and is almost independent of temperature, in contrast to the experiTable 4 Theoretical and experimental values of the kinetic isotope effect

kF+HD/kF+DH T (K)

159 168 222 298 413

A H / A De ,.~Ea t

Experiment a

VTST

QCT

T5Ab

T5A c

M5 d

6SEC c

1.55 1.55 1.48 1.45 1.33

1.60 1.58 1.50 1.44 1.39

1.04 1.05 1.05 1.05 1.05

1.09 1.08 1.02 0.97 0.94

1.36 1.35 1.31 1.27 1.25

1.26 -0.07

1.27 -0.07

1.06 +0.01

0.85 -0.08

1.17 -0.05

a Persky [4]. b Results of variational transition state theory reported by Steckler et al. [26]. c This work. d Values calculated from an Arrhenius fit to the rate constants reported by Muckerman [19] for the temperature range 1 5 0 - 4 5 0 K. ¢ Ratio of pre-exponential factors for F + H D and F + DH. f Activation energy difference Ea(F + H D ) - E,(F + DH) in kcal/mol.

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

mental values and trend. For the M5 surface, the trend is in the correct direction, but the absolute values are very low. They are even lower than unity at temperatures above 222 K (formation of DF is preferred over the formation of HF). The VTST results for the T5A surface agree very well with experiment, but differ considerably from the QCT results on the same surface.

3. Energy partitioning and vibrational branching ratios of products The energy partitioning and the vibrational state distributions among reaction products were calculated for thermal distributions of collision energies and of initial rotational states at 300 K. Results for

301

the reactions F + H 2, F + D 2 , F + HD and F + DH, for the potential energy surfaces TSA and 6SEC, are compared with the experimental results of Perry and Polanyi [8] in Table 5. Experimental results for fv and for the vibrational branching ratios of the products reported by Berry [6] for the four isotopic reactions, and by Chang and Setser [7] for F + H 2 , are in good agreement with the results of Perry and Polanyi. It can be seen that for both surfaces the fraction of available energy appearing as vibrational energy fv is higher than the experimental value, and the fraction appearing as translational energy fT, is lower than the experimental value for all the isotopic reactions, but the results for the 6SEC surface are closer to the experimental results. Actually, for F + H 2 and F + D 2 f v and fT calculated for the 6SEC surface are not far from the experimental values and

Table 5 Theoretical and experimental energy partitioning and relative vibrational state distributions of products at 300 K F+H2~HF+H

F+D2~DF+D

F+HD~HF+D

F+DH~DF+H

( fv )a ( fR ) b (fr) c

0.79 0.05 0.16

0.79 0.04 0.17

0.79 0.06 0.15

0.80 0.03 0.17

k ( t ' = 1) d

0.07

--

0.08

--

k( c' = 2) k ( t ' = 3) k(c' = 4)

1.00 1.36 -

0.16 1.00 1.30

1.00 1.24 -

0.09 1.00 1.42

0.73 0.07 0.20

0.73 0.06 0.21

0.73 0.10 0.17

0.73 0.06 0.21

0.13 1.00 0.81 -

0.44 1.00 0.77

0.14 1.00 0.72 -

0.35 1.00 0.90

0.664 0.083 0.260

0.665 0.076 0.259

0.588 0.125 0.287

0.626 0.066 0.308

0.28 1.00 0.55 -

0.15 0.52 1.00 0.59

0.30 1.00 0.14 -

0.18 0.54 1.00 0.61

T5A-QCT

e

6SEC-QCT e ( fv ) ( fR )

(fT) k(v'= k(v' k(v' = k(L/ =

1) 2) 3) 4)

Experimental f ( fv ) ( fR ) ( fy ) k(c,' = k( e,' = k(t" k(L" =

1) 2) 3) 4)

a Fraction of available energy appearing as vibrational energy of the products. b Fraction of available energy appearing as rotational energy of the products.

Fraction of available energy appearing as translational energy of the products. d Relative rate constant for the fraction of products in the vibrational state t ' .

e This work. f Perry and Polanyi [8].

302

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303

the fractions appearing as rotational energy fR agree quite well with the experimental results. The experimental results of Perry and Polanyi concerning the vibrational state distributions of the products indicate that the most populated state for the reactions F + H 2 ~ HF + H and F + HD ~ HF + D is H F ( v ' = 2 ) , and for F + D 2 ~ D F + D and F + DH ~ DF + H it is DF(v' = 3). The molecular beam experiments of Neumark et al. [10,11] for specific collision energies, show the same qualitative behavior. The QCT results in Table 5 also indicate the same qualitative behavior for the 6SEC surface, but not for the T5A surface, for which it was found that the most populated states are H F ( u ' = 3) and DF(v' = 4). Though the vibrational branching ratios HF(v' = 3)/HF(v' = 2) and DF (v' = 4 ) / D F (v' = 3) calculated for the 6SEC surface are lower than unity, in qualitative agreement with experiment, they are too high, and the ratios HF(v' = 1)/HF(v' = 2) and D F ( v ' = 2 ) / D F ( v ' = 3) are too low. Qualitatively similar QCT results for this surface were obtained by Aoiz et al. [36] for specific collision energies. In contrast, OCT calculations for the new SW surface [18] systematically gave too low values for HF(v' = 3)/HF(v' = 2 ) and too high values for H F ( v ' = 1)/HF(v' = 2).

4. Conclusions In this paper we report results of QCT calculations of kinetic data and of the energy partitioning among reaction products under thermal conditions, for the F + H 2 reaction and its isotopic analogs, for the potential energy surfaces T5A and 6SEC. The results are compared with QCT results for the M5 surface and with experimental data. The main conclusion from this study is that, in general, the 6SEC surface reproduces qualitatively most of the experimental results. The agreement with experiment is much better than for the two other surfaces. This conclusion is consistent with conclusions of Aoiz et al. which are based mainly on calculations of vibrationally state resolved integral and differential cross sections and of vibrational branching ratios for specific collision energies. The kinetic results for the 6SEC surface, except for the values of the kinetic isotope effect kF+laz/kF+D2 which are much lower than the exper-

imental values, are in reasonable agreement with experiment. The reactivity which was found to be higher for the 6SEC surface than for the M5 and T5A surfaces, and which brings the calculated rate constants kv+ H2 and kv+ D2 closer to the experimental values, can probably be attributed to the nonlinear configuration of the transition state and to the flat bending potential for this surface. Because of these features, collisions with wider ranges of impact parameters and of F - H - H angles than for the other surfaces are reactive, leading to larger cross sections and larger rate constants and also to more forward scattering. The isotopic branching ratios OR(F + HD)/O'R(F + DH) calculated for the 6SEC surface exhibit the same qualitative behavior as the experimental ratios, though the absolute values are lower. They are lower especially at the higher collision energy range, but approach the experimental values at the lower energy range, which is the more important range for calculations of rate constants at room temperature. This is confirmed by the fact that for a thermal distribution of collision energies at 300 K the calculated ratio (the intramolecular kinetic isotope effect) is in good agreement with the experimental value. The energy partitioning and vibrational branching ratios calculated for the 6SEC surface, for the F + H 2 reactions and its isotopic analogs, are in qualitative agreement with experiment. This is especially true for F + H 2 and F + D 2. For these reactions the values of fR are in good agreement with the experimental values, and the values of fv and fT are also not much different from the experimental values. A larger difference is found for the vibrational branching ratios. An interesting observation from the present study of the 6SEC surface is the non-monotonic behavior of the excitation functions for F + H 2 (v = 0, j = 0) and F + D 2 ( v = 0 , j = 0 ) for collision energies above approximately 3 kcal/mol. This behavior is observed for total cross sections and becomes more pronounced for vibrationally state resolved cross sections for HF ( v ' = 2) and DF ( v ' = 3). It would be very useful to test these predictions by molecular beam measurements of the vibrationally state resolved excitation functions for F + p-H 2 at low temperatures. It is also desirable to have results of QM calculations for the 6SEC surface for the quantities dis-

E. Rosenman, A. Persky / Chemical Physics 195 (1995) 291-303 c u s s e d here a n d to c o m p a r e t h e m w i t h the Q C T results. S u c h c a l c u l a t i o n s will i n d i c a t e w h e t h e r Q M effects, s u c h as t u n n e l i n g , are i m p o r t a n t for this p o t e n t i a l e n e r g y surface. If this is the case, t h e n o n e m a y e x p e c t larger t u n n e l i n g c o r r e c t i o n s for F + H 2 t h a n for F + D 2, a n d for F + H D t h a n for F + D H , as n e e d e d in o r d e r to n a r r o w the g a p b e t w e e n calcul a t i o n s a n d e x p e r i m e n t . It w o u l d also b e v e r y h e l p f u l to carry out s i m i l a r Q C T a n d Q M c a l c u l a t i o n s for the n e w S W surface. T h e i n f o r m a t i o n f r o m s u c h studies for t h e s e t w o surfaces, w h i c h are a p p a r e n t l y the b e s t s u r f a c e s e x i s t i n g at p r e s e n t , will b e v e r y h e l p f u l in a s s e s s i n g t h e i r a c c u r a c y a n d the i m p r o v e m e n t s that are still n e e d e d .

Acknowledgement W e are g r a t e f u l to P r o f e s s o r D.G. T r u h l a r for p r o v i d i n g us w i t h the c o m p u t e r p r o g r a m s for the T 5 A a n d 6 S E C p o t e n t i a l e n e r g y surfaces.

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