The unimolecular dissociation of the propionyl radical ...

The unimolecular dissociation of the propionyl radical: A classical dynamics study Emilio Martı́nez-Núñez, Angeles Peña-Gallego, and Saulo A. Vázquez Citation: The Journal of Chemical Physics 114, 3546 (2001); doi: 10.1063/1.1322628 View online: http://dx.doi.org/10.1063/1.1322628 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/114/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Direct-dynamics VTST study of hydrogen or deuterium abstraction and C–C bond formation or dissociation in the reactions of CH3 + CH4, CH3 + CD4, CH3D + CD3, CH3CH3 + H, and CH3CD3 + D J. Chem. Phys. 138, 194305 (2013); 10.1063/1.4803862 Comprehensive theoretical studies on the CF3H dissociation mechanism and the reactions of CF3H with OH and H free radicals J. Chem. Phys. 126, 034307 (2007); 10.1063/1.2426336 Quasiperiodic trajectories in the unimolecular dissociation of ethyl radicals by time-frequency analysis J. Chem. Phys. 123, 021101 (2005); 10.1063/1.1950673 Dissociation dynamics of the Ã 2 A ″ state of vinyl radical J. Chem. Phys. 118, 4452 (2003); 10.1063/1.1542878 The unimolecular dissociation of HCO. V. Mixings between resonance states J. Chem. Phys. 115, 8876 (2001); 10.1063/1.1412601

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 114, NUMBER 8

22 FEBRUARY 2001

The unimolecular dissociation of the propionyl radical: A classical dynamics study Emilio Martıńez-Nuń˜ez, Angeles Penã-Gallego, and Saulo A. Va´zqueza) Departamento de Quı´mica Fı´sica, Universidad de Santiago de Compostela, Santiago de Compostela E-15706, Spain

共Received 2 May 2000; accepted 14 September 2000兲 The unimolecular dissociation of the propionyl radical to form CO and CH2CH3 was investigated by classical trajectory calculations. Various types of initial sampling conditions were employed: Microcanonical for energies ranging from 27.8 to 72.8 kcal/mol above the zero-point energy 共ZPE兲, and selective excitations at 67.8 kcal/mol. A quasiclassical barrier sampling technique, which circumvents the problem of ZPE leakage, was also used for the calculation of product energy distributions. For energies above 43 kcal/mol, the computations showed that the intramolecular vibrational relaxation is not rapid as compared with the rate of reaction. On the other hand, it is found that vibrational modes associated to the CCO moiety are significantly coupled to the reaction coordinate, in agreement with the suggestion reported by Zewail and co-workers 关J. Phys. Chem. 100, 9202 共1996兲兴. However, the calculations cannot predict the significant decrease of the dissociation rate observed upon deuterium substitution on the ␣-carbon. Product energy distributions and CO vibrational populations computed for the different excitation schemes are compared with those determined experimentally. For many ensembles, the fraction of the available internal energy resulting in CO vibration agrees with that estimated experimentally. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1322628兴

I. INTRODUCTION

rates and found that the experimental rate decreases by a factor of ⬃2 by deuterium substitution on the ␣-carbon but for ␤-D substitution the rate is almost unaffected.4 On the contrary, their RRKM predictions yield k H /k D ratios slightly above unity for both substitutions. They suggested that modes associated to the motion of ␣-D atoms are primarily involved in the dynamics, while modes associated to the motion of ␤-D atoms have little involvement due to a slow intramolecular vibrational redistribution 共IVR兲. Using time-resolved Fourier transform emission spectroscopy, Hall et al.7 studied the fragmentation of diethyl ketone and the subsequent secondary dissociation of the propionyl radical at 67.8 kcal/mol above the ZPE. The authors determined the product energy distributions 共PEDs兲 in the fragments 共CO and ethyl radical兲 and found that both a fully statistical model and an impulsive model could not predict the CO vibrational distribution. They suggested that this result may arise from an incomplete IVR in the energized propionyl radical. In addition, they concluded that, surprisingly, the propionyl radical appears to approach the complete IVR limit less closely than does the acetyl radical. This contrasts with the result of Kim et al.4 indicated above. The present paper is focused on the study of the dissociation dynamics of the propionyl radical by means of classical trajectory simulations. In the first part of this paper, we describe the potential energy surface 共PES兲 used in the trajectory calculations. On account of the size of our system, we have employed a formalism based on the use of internal valence coordinates, which has been successfully applied to other polyatomics.1,9–11 Our semiempirical surface conforms with the experimental barrier height and endothermicity,7,12

In our recent classical dynamics studies of the unimolecular dissociation of the acetyl radical,1,2 we concluded that the dissociation of the radical is intrinsically non-RRKM3 at 65 kcal/mol and that mode specific effects are clearly exhibited, in agreement with femtosecond mass spectrometric measurements.4 However, at low energies 共below 41 kcal/ mol兲, the dynamics was shown to be intrinsically RRKM1 and the calculated rate constants in accord with those obtained by ultrafast deep ultraviolet 共UV兲 mass-resolved photoionization spectroscopy.5 Near the threshold energy, we found2 that the rotational degrees of freedom are almost adiabatic and the microcanonical rate constants, corrected to take the zero-point energy 共ZPE兲 leakage into account, are substantially higher than the experimental estimations determined by Shibata et al.6 The differences between the theoretical and the experimental rates were attributed to a nonrandom initial distribution of the energy,2 which may cause apparent non-RRKM behavior.3 Kim et al.4 studied the unimolecular dissociation of the propionyl and acetyl radicals as well as its ␣-D and ␤-D isotopic analogues by means of fs-resolved mass spectrometry at 65⫾10 kcal/mol of internal energy. They obtained rates that substantially differ from their RRKM predictions, although the discrepancy was less pronounced for the propionyl than for the acetyl radical. They concluded that the energy partitioning in the propionyl and acetyl radicals could be nonrandom, thus leading to nonstatistical behavior. They also studied the effect of selective isotopic labeling on the a兲

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0021-9606/2001/114(8)/3546/8/$18.00

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© 2001 American Institute of Physics


J. Chem. Phys., Vol. 114, No. 8, 22 February 2001

Dissociation of the propionyl

rather than with the corresponding ab initio data,8 in order to evaluate reliable PEDs and compare them with those obtained experimentally.7 The PES was used to perform classical dynamics calculations at energies ranging from 27.8 to 72.8 kcal/mol above the zero-point energy, thus covering the energies employed in the aforementioned experimental studies.4,7 Several nonrandom initial distributions of energy 共including rotational and normal/local mode excitations兲 were selected to explore the possibility of non-RRKM effects and the rate constants and PEDs were compared with those observed experimentally.7 In addition, the energy dependence of the microcanonical rates was fitted to several models incorporating anharmonicity, yielding better results than the commonly used classical harmonic RRKM scheme.

II. POTENTIAL ENERGY SURFACE

The potential function employed in this work is based on that used previously for the acetyl radical,1 appropriately modified to include the additional terms for the propionyl radical. Particularly, we substituted a C–H stretch interaction by the corresponding interactions in the C–CH3 moiety. The general expression reads V⫽V 共 R CO兲 ⫹V 共 R CC␣ 兲 ⫹V 共 R C␣ C␤ 兲 2

⫹

兺

i⫽1

3

V 共 R C␣ H共i) 兲 ⫹

兺 V 共 R C H共i) 兲 ⫹V 共 ␪ CCO兲 ␤

i⫽1

2

⫹

兺

i⫽1

3

V 共 ␪ CC␣ H共i) 兲 ⫹V 共 ␪ CCC兲 ⫹

3

⫹

兺兺

i⫽1 j⫽i⫹1

V 共 ␪ H共i兲C␤ H共j) 兲 ⫹V 共 ␻ C␤ C␣ HH兲 3

兺 V 共 ␶ OCCH共i兲 )⫹V 共 ␶ OCCC兲 ⫹ i⫽1 兺 V 共 ␶ C C H共i) 兲

III. DYNAMICS STUDY

3

兺兺

i⫽1 j⫽1

V 共 ␶ H共i兲C␣ C␤ H共j) 兲 ⫹V 共 R CC␣ , ␪ CCO兲 .

共1兲

Hereinafter, the subscripts ␣ and ␤ will be used when necessary to label the ␣-carbon and ␤-carbon, respectively. Following is a description of the changes made in the acetyl PES 共described in Ref. 1兲 as well as the new terms included here to construct the propionyl PES. For the CC␣ 共here C stands for the carbonyl carbon兲 stretching interaction 关Eq. 共5兲 in Ref. 1兴, the potential well depth was modified according to p p r ⫺ 共 D CC ⫺D CC D CC␣ ⫽D CC 兲 SW共 6 兲 , ␣ ␣ ␣

共2兲

p r where D CC and D CC are the potential well depths in the ␣ ␣ product and reactant regions, respectively, and SW共6兲 is a switching function defined in Ref. 1. The potential term for the new C␣ C␤ stretch adopted the same form of Eq. 共5兲 in Ref. 1, but with the bond length parameter varied by SW共1兲

R C␣ C␤ ⫽R Cp

where, as above, the superscript r or p refers to the reactant or the product, respectively. Bends CCC and CC␣ H were modeled by Eq. 共9兲 of Ref. 1, and C␣ C␤ H and HC␣ H by Eq. 共10兲 of that reference. The wagging interaction in the product CH3CH2 was modeled by Eq. 共11兲 of Ref. 1. Finally, the new torsional interactions adopted the same expression as those for the acetyl radical 关Eq. 共13兲 in Ref. 1兴. Table I collects the main geometrical features, frequencies, and relative energies taken from the literature,1,7,12,13 and also the corresponding values obtained by our model PES. The parameters of our potential function 共given in Table I in PAPS兲14 were taken from ab initio calculations8 or adjusted to reproduce the vibrational frequencies and relative energies. In particular the equilibrium bond lengths and bond angles were taken from our previously reported B3LYP/6-31G(2d,2p) calculations8 whereas force constants, curvature parameters, and potential well depths were taken from the corresponding model potential for the acetyl radical.1 Finally, C i parameters were adjusted to model the experimental energetics and the torsional parameters recalibrated to give the correct methyl and ethyl orientations. As far as geometries are concerned, there is good agreement between the ab initio and our model PES 共see Table I兲, because we used the ab initio values as parameters in the potential function. For most of the vibrational frequencies, the agreement is reasonable. To a certain extent, the discrepancies for some frequencies may come from the fact that, as mentioned in the Introduction, we wanted our PES to match the experimental barrier height7,12 and endothermicity, which differ substantially from the ab initio values. For example, the experimental barrier height is 17.5 kcal/mol 共17.3 kcal/ mol with our model PES兲 whereas the CBS-QB3 value8 is 14.9 kcal/mol. The reverse barrier calculated with out PES 共12.9 kcal/mol兲 conforms with the experimental estimate 共13 kcal/mol兲.7,12

␣ ␤

i⫽1 2

⫹

␣ ␤

3

2

⫹

兺 V 共 ␪ C C H共i) 兲

i⫽1

3547

␣ C␤

⫺ 共 R Cp

␣ C␤

⫺R Cr

␣ C␤

兲 SW共 1 兲 ,

共3兲

A. Trajectory computational details

We have evaluated rate constants for the propionyl radical decomposition at several energies between 27.8 and 72.8 kcal/mol above the ZPE of the reactant 共44.24 kcal/mol兲 by using the classical trajectory methodology as implemented in the GENDYN code.15 We used the efficient microcanonical sampling 共EMS兲16 technique, which appropriately accounts for the anharmonicity of the potential in the selection of the initial conditions, to prepare a microcanonical ensemble of reactant molecules. Details about the Markov chain generation are reported in Ref. 1. To explore the possibility of mode specificity, normal or local mode excitations were used as in previous work.1,2,9–11 It is well-known that classical trajectories may not give accurate PEDs due to the improper treatment of ZPE. For this reason, we additionally used a quasiclassical barrier sampling 共QCBS兲 technique,17 which initiates the trajectories at the barrier and populates the vibrational modes in accord with quantum RRKM theory. The method is based on the


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Martıńez-Nuń˜ez, Penã-Gallego, and Va´zquez


TABLE I. Main geometrical features,a energetics, and frequencies for the chemical species involved in this study. Internal coordinate CO CC␣ C␣C␤ CCO

syn-CH3CH2CO b

PES

1.180 1.520 1.530 126.8 184 217 374 571 810 836 951 1015 1021 1153 1402 1506 1525 1548 1551 1915 2795 2838 2905 2909 2922 0.0

ab initio

1.180 1.520 1.530 126.8 108 233 241 627 733 793 970 1049 1084 1273 1319 1417 1454 1500 1505 1916 3046 3048 3069 3110 3119 0.0

CH3CH2⫹CO

TS c

PES

b

c

ab initio

PES

b

ab initioc

Expt.d

1.139

1.128

1.141 1.140 1.139 2.259 2.238 1.498 1.496 111.6 113.7 Harmonic frequencies/cm⫺1 165i 282i 70 28 45 116 147 235 151 357 322 779 548 868 798 864 879 825 886 964 1013 947 1072 1065 1082 1246 1216 1325 1481 1407 1479 1541 1476 1537 1541 1486 1537 1721 1492 1646 1827 2067 1827 2795 2985 2795 2822 3061 2817 2908 3099 2909 2909 3132 2909 2922 3220 2926 Relative energiese/共kcal/mol兲 19.8共17.3兲 14.9 17.7共12.9兲

13

a

Distances in Å and angles in degrees. This study. c B3LYP/6-31G(2d,2p) geometries and frequencies, and CBS-QB3 energies from Ref. 8. d Taken from Refs. 7 and 12. e Values in parentheses are corrected to take the ZPEs into account. b

classical barrier sampling18 and selects each state with equal probability so that the momentum distribution in the reaction coordinate agrees with RRKM theory.19 Details of the sampling procedure has been reported elsewhere17 and here we present a concise description only. The probability of assigning n i quanta to a normal mode 共orthogonal to the reaction coordinate兲 is given by P 共 n i 兲 ⫽W ni /W tot ,

共4兲

where W tot⫽兺i⫽0Wni and W ni is the number of quasiclassical states with energy less than the microcanonical trajectory energy (E traj) and with mode i having n i quanta. W ni can be calculated by the Beyer–Swinehart algorithm.20 The procedure is repeated for each normal mode such that the available m⫺1 energy in normal mode m is ⭐E traj⫺ 兺 i⫽1 (n i ⫹ 21 )ប ␻ i . Once the energy in each normal mode has been selected, the Cartesian coordinates and momenta are obtained from the displacement of the normal modes from the equilibrium at the transition state in the traditional way.21 Additionally, some of the ensembles studied in this work incorporated rotational excitation. To select initial conditions for rotating propionyl radical, we put it into the principal axes of inertia frame and then excited rotational motion around one of the axes. Particularly, we employed a total

rotational energy of 10 kcal/mol in these ensembles, keeping the total energy constant. Thus, 57.8 kcal/mol were assigned for vibrational excitation and 10 kcal/mol for a axis rotation or for b and c axes rotations 共5 kcal/mol for each axis兲. For each kind of initial conditions, the lifetime of a trajectory was taken to be the integration time up to the last inner turning point of the C␣ C vibration. The trajectories were integrated by using the Runge–Kutta–Gill routine as implemented in the GENDYN code with a step size of 0.08 fs. This value was selected on the basis of a trial-and-error procedure so that each individual trajectory arrived at the same result with a smaller step size. Additionally, an average energy conservation of five significant figures was observed. Batches of 3000 trajectories were integrated for a maximum of 30 ps or until dissociation occurred 共more specifically, until the C␣ C interatomic distance exceeded 10 Å兲. With these conditions, more than 90% of the trajectories dissociated at all the energies studied here. When a trajectory finished, a final product analysis was performed and various classical mechanical molecular properties, such as relative translational energy, product vibrational, and rotational energies, were calculated from the atomic Cartesian coordinates and momenta.22,23 In addition, vibrational quantum numbers




TABLE II. Energy dependence of the trajectory rate constants.a E/共kcal/mol兲 27.8 32.8 37.8 42.8 47.8 52.8 57.8 62.8 65⫾10 67.8 72.8 a

k RRKM/ps⫺1

k/ps⫺1

0.209⫾0.003 0.298⫾0.006 0.426⫾0.007 0.559⫾0.008 0.722⫾0.012 0.897⫾0.014 1.151⫾0.020 1.319⫾0.018

0.215⫾0.004 0.305⫾0.009 0.443⫾0.010 0.579⫾0.012 0.757⫾0.016 0.937⫾0.020 1.240⫾0.028 1.403⫾0.026 1.818; 1.333b

1.500⫾0.018 1.777⫾0.026

1.632⫾0.026 1.909⫾0.035

Energy above the ZPE. Rate constants obtained by fits to Eq. 共5兲. Experimental rate constants at 65 kcal/mol taken from Ref. 4. The results correspond to dissociation of the propionyl radical prepared from methyl ethyl ketone and diethyl ketone, respectively.

b

for the product CO were evaluated by the Einstein– Brillouin–Keller 共EBK兲24–26 quantization of the action integral.

The dependence of the microcanonical rates with energy can help us to determine the anharmonicity of our potential energy surface at the energies of interest. We have used various models based on previous work27,28 to predict this energy dependence. Model 1 is the well-known classical harmonic RRKM20 approach k 共 E 兲 ⫽a 1

冉

E⫺E 0 E

冊

s⫺1

,

共6兲

where a 1 is an adjustable frequency factor 共the subscript 1 comes from model 1兲 and E 0 and s are the classical barrier height and the number of degrees of freedom, respectively. The value of E 0 was taken to be 19.8 kcal/mol. In models 2 and 3, the lowest frequencies at the transition state are treated as free rotors instead of harmonic oscillators. It is straightforward to show27 that in the case of one rotor at the transition state, with the other modes treated as harmonic oscillators 共model 2兲 k 共 E 兲 ⫽a 2

共 E⫺E 0 兲 s⫺3/2 , E s⫺1

共7兲

and for two rotors at the transition state 共model 3兲

B. Results and discussion

k 共 E 兲 ⫽a i

1. EMS initialization

Classical anharmonic RRKM rate constants were calculated from our trajectory simulations at energies varying from 27.8 to 72.8 kcal/mol above the ZPE. The rates were extracted from the t⫽0 intercept of each of the lifetime distributions19 P(t)⫽⫺1/N(0)dN(t)/dt. For all the energies, the lifetime distributions 共see Fig. 1 in PAPS兲14 were satisfactorily fitted to the expression: P 共 t 兲 ⫽k RRKM⫻exp共 ⫺kt 兲 .

3549

共5兲

Table II shows the RRKM rates and the kinetic parameter k. As seen from the table, at the lowest energies 共up to 42.8 kcal/mol兲, k RRKM and k agree within statistical errors, suggesting that the intramolecular dynamics is ergodic on the time scale of the unimolecular decomposition.3,19 As the energy rises, these two parameters no longer agree, although the difference between both values is not very large. This situation may arise when IVR is not rapid enough as compared with the rate of reaction.3,19 Similar results were previously found in the unimolecular dissociation of the acetyl radical.1 Kim et al.4 reported theoretical RRKM and experimental rate constants at 65⫾10 kcal/mol. They found that the discrepancy between the RRKM and experimental rates is less pronounced for the propionyl than for the acetyl radical. The experimental rates for the fragmentation of the propionyl radical prepared from methyl ethyl ketone and diethyl ketone are 1.818 and 1.333 ps⫺1, respectively. These values seem to be in reasonable agreement with ours taking into account the experimental uncertainty in the excitation energy. However, a nonrandom initial preparation of the molecules is expected to occur in the fs-resolved mass spectrometry study,4 and therefore, a direct comparison with our results must be taken with care because mode specific effects could be exhibited. This will be the subject of the next section of this paper.

共 E⫺E 0 兲 s⫺2

E s⫺1

,

共8兲

where the adjustable parameters a 2 and a 3 now include, among other constants, the rotational constants of the rotors and, therefore, have different units. Following previous work,27,28 we have used two additional models in which the frequency factor of Eq. 共6兲 is now energy dependent: a⫽a 4

exp共 b ts共 E⫺E 0 兲兲 , exp共 bE 兲共 1⫹bE/s 兲

共9兲

In model 4, Eq. 共9兲 is inserted into Eq. 共6兲 with a instead of a 1 . Model 5 is the same as model 4 but with a power dependence in (s⫺2) instead of (s⫺1) in the numerator of Eq. 共6兲, as corresponds to consider the two lowest frequencies as free rotors. The least square fits of the above expressions to our microcanonical trajectory rates yielded the curves plotted in Fig. 1 and the parameters listed in Table III. As seen from the figure, models 4 and 5 describe accurately the energy dependence of the rates. For model 4, the anharmonicity correction factor of the reactant density of states 关the denominator of Eq. 共9兲兴 evaluated at the critical energy was calculated to be 1.25. Although this value is very low, it increases significantly as the energy rises so that at the energies selected here one may expect that anharmonic effects play a significant role in the dynamics of the propionyl radical. In fact, the usual classical harmonic RRKM expression 共model 1兲 does not accurately fit our rates as seen in Fig. 1 共dotted line兲. However, when the two lowest frequency vibrations at the transition state are considered as free rotors 共model 3兲, the energy dependence of the microcanonical trajectory rates can be modeled quite accurately 关see Fig. 1 and the rootmean-square 共rms兲 deviations in Table III兴. These results suggest that to a large extent anharmonicity comes from the


3550



case, each ensemble is named by the internal coordinate dominating the normal mode that is initially excited. Additionally, for ensembles EMS, ␣-H, and ␤-H, rotational excitation was considered, as explained in Sec. III A, by exciting rotation around one of the three principal axes of inertia. In an obvious notation, EMSa represents an ensemble in which rotational excitation is on the a axis, EMSbc represent b and c axes rotations, and so on. As indicated above, a total energy of 10 kcal/mol was assigned in each rotational excitation scheme. The lifetime distributions obtained for the above ensembles were fitted either to a single-exponential P 共 t 兲 ⫽k 1 exp共 ⫺k 2 t 兲 , FIG. 1. Microcanonical rates and fits by the models considered in this work.

共10兲

where k 1 and k 2 are the adjustable kinetic parameters, or to a double-exponential

lowest frequency vibrations at the transition state. As we will show later on and we have previously found for the acetyl radical1 and methyl nitrite,27 the treatment of these degrees of freedom is also crucial for an appropriate classical interpretation of the isotope effects.

2

P共 t 兲⫽

兺 c i 兵 k i exp共 ⫺k i t 兲其 ,

共11兲

i⫽1

where c i represents the percentage of trajectories decaying with a certain rate k i . The parameters resulting from the fits to one of the above expressions are collected in Table IV. Regardless the pattern of initial excitation, the RRKM model requires the phase space distribution to become random in a negligible short time. However, this is not our case. For example, for ensembles CCO, CC␣ , and CO, a significant percentage of trajectories decay with a rate one order of magnitude faster than the microcanonical one. If there were strong internal couplings, the lifetime distribution will become that of the RRKM theory after rapid IVR. However, for the above ensembles only k 2 in Eq. 共11兲, which amounts 1.5–1.6 ps⫺1, compares well with the RRKM rate at 67.8 kcal/mol 共1.50 ps⫺1兲. From a general inspection of the rate constants of Table IV, we can conclude that the normal modes of the CCO moiety and the C␣ – H local modes are strongly coupled to the reaction coordinate whereas the C␤ – H local modes are only modestly coupled. This result could be expected from the normal mode eigenvectors and/or the distance to the reaction coordinate mode within the molecule. Thus, a C␣ – H local mode should enhance the reaction rate more than does a C␤ – H local mode, as predicted by our calculations and suggested by Kim et al.4 Our calculations also predict that the dissociation rates diminish when the energy is partitioned into vibration and rotation. This result is in line with those obtained for the acetyl radical,2 wherein we found that the rotational degrees

2. Nonrandom initialization

In this work, several ensembles were selected at a total energy of 67.8 kcal/mol above the ZPE to compare with the results of two experiments: the fs-resolved mass spectrometry study of Kim et al.,4 in which the propionyl radical contains 65⫾10 kcal/mol of internal energy, and the timeresolved Fourier transform emission spectroscopic study of Hall et al.,7 carried out at 67.8 kcal/mol above the propionyl ground state. While the results of the latter will be used for a comparison of the PEDs, the former provided rate constants that can be compared with those reported in this section. Following is a description of the ensembles considered here at the above energy. EMS represents a microcanonical distribution of the energy as in the previous section. For the remaining ensembles the normal/local mode initially excited gives the name of the ensemble. Therefore, in ␣-H the two C␣ – H stretches associated to the methylene group were excited above their ZPEs according to the local mode approach, with the remaining modes containing only their ZPEs. Similarly, ensemble ␤-H corresponds to the initial excitation of the three C␤ – H stretches of the methyl group. The normal mode excitation scheme was used for ensembles denoted as CCC, CCO, C–C␣ , and CO, which involve the excitation of the vibrational modes with wave numbers 217, 571, 1402, and 1915 cm⫺1, respectively 共see Table I兲. In this

TABLE III. Parameters of the least square fits of the microcanonical trajectory rates to several RRKM models. Model

a ia

b

b ts

rmsb

1 2 3 4 5

77.8 ps⫺1 735.0 ps⫺1共kcal/mol兲0.5 6877.2 ps⫺1共kcal/mol兲 253.675 ps⫺1 6037.23 ps⫺1共kcal/mol兲

¯ ¯ ¯ 1.1⫻10⫺2 (kcal/mol) ⫺1 9⫻10⫺6 (kcal/mol) ⫺1

¯ ¯ ¯ 8.0⫻10⫺4 (kcal/mol) ⫺1 1.5⫻10⫺3 (kcal/mol) ⫺1

0.10 0.05 0.026 0.023 0.019

a

The subscript i stands for model i. Root-mean-square deviation. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: b

193.144.81.13 On: Fri, 21 Nov 2014 07:48:51



TABLE IV. Trajectory and experimental rate constantsa 共in ps⫺1兲 for the ensembles considered in this work at 67.8 kcal/mol above the ZPE. Ensemble

Type of exponential

b

Expt. EMS EMSa EMSbc ␣-H ␣-Ha ␣-Hbc ␤-H ␤-Ha ␤-Hbc CCC CCO CC␣ CO

k1

k2

1.818; 1.333 1.500⫾0.0018 1.632⫾0.026 1.480⫾0.027 1.598⫾0.038 1.359⫾0.014 1.448⫾0.020 2.299⫾0.038 2.062⫾0.036 1.692⫾0.028 1.571⫾0.028 2.042⫾0.026 1.853⫾0.024 1.529⫾0.048 1.100⫾0.030 1.483⫾0.046 1.003⫾0.025 1.444⫾0.044 1.017⫾0.026 2.105⫾0.040 1.746⫾0.031 9.01⫾0.40 共81%兲 1.50⫾0.04 共19%兲 8.16⫾0.70 共25%兲 1.47⫾0.06 共75%兲 14.15⫾0.32 共91%兲 1.62⫾0.05 共9%兲

single single single single single single single single single single double double double

The parameters k 1 and k 2 were obtained from fits to single- 关Eq. 共10兲兴 or double- 关Eq. 共11兲兴 exponentials. For the double-exponential fits, the percentages of trajectories decaying with a particular rate (c i ) are indicated in parentheses. Quoted errors correspond to a 95% confidence limit interval. b Taken from Ref. 4. The results correspond to dissociation of the propionyl radical prepared from methyl ethyl ketone 共1.818 ps⫺1兲 and diethyl ketone 共1.333 ps⫺1兲. a

of freedom behave adiabatically. However, the study of the acetyl radical was carried out at energies near the threshold and, therefore, the comparison should be taken with caution. On the other hand, in the present study we observe rotational specificity for ␣-H excitation: The rates for rotation around the b and c axes are greater than those obtained for a axis rotation. At a first sight, it seems that this result is a consequence of the centrifugal force, which should favor dissociation when rotation occurs around the b or c axis. However, this trend is not observed for the EMS and ␤-H initializations. For a further comparison with experiment, we analyzed the product rotational, vibrational, and translational energies for our different patterns of internal energy distribution at a total energy of 67.8 kcal/mol above the ZPE. In addition, due to the improper treatment of ZPE by classical trajectories we

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have selected an additional ensemble 共referred to as QCBS兲 in which the trajectories are initiated at the barrier according to a quantum RRKM distribution of energy as mentioned above. The PEDs obtained for our ensembles are gathered in Table V and compared with the experimental data.7 The results obtained for ensemble QCBS are substantially different from those calculated under EMS initial conditions. In particular, the fractions of the available energy resulting in CH2CH3 vibration are 0.56 and 0.72 for the QCBS and EMS ensembles, respectively. To a large extent, this disagreement may arise from the ZPE leakage in the EMS trajectories. The substantial differences for the PEDs calculated for the other ensembles indicate a clear specificity of modes. On the other hand, for many ensembles, the fraction of the available energy resulting in CO vibrational energy is in good accord with the experimental value 共0.04兲.7 The vibrational energy content of the ethyl radical, however, is in general larger in this study than in the experimental investigation.7 However, the experimental quantity was not obtained directly, rather Hall et al.7 combined the experimental value for the vibrational energy content of CO with ab initio calculations to obtain the Einstein coefficients for the five C–H stretching modes. As a consequence, their value could be affected by some errors. The CO vibrational quantum number distributions for the ensembles described above were also calculated 共see Table II in PAPS兲.14 All the distributions peak at v ⫽0 with the experimental one being, in general, broader than those obtained here. An exception occurs for the distributions obtained under initial excitation of modes 571 共CCO ensemble兲 and 1915 共CO ensemble兲, which show that a substantial number of trajectories involved high vibrational quantum numbers. We therefore suggest that modes involving the motion of the CCO moiety could be initially excited in the experiment of Hall et al.7 Additionally, the QCBS ensemble predicts substantially different vibrational distributions 共mainly those for v ⭓1兲 than those predicted by the EMS ensemble. This fact suggests, again, that a significant ZPE leakage affects the trajectory results.

TABLE V. Product energy distributions obtained experimentallya and from our classical trajectory calculations at 67.8 kcal/mol. Ensemble

f 具 Etr典

f 具 Er, CH2CH3典

f 具 Ev, CH2CH3典

f 具 Er, CO典

f 具 Ev, CO典

0.23 0.13 0.13 0.19 0.11 0.12 0.18 0.11 0.12 0.18 0.19 0.22 0.11 0.11

0.08 0.06 0.11 0.08 0.04 0.11 0.03 0.05 0.11 0.04 0.07 0.05 0.04 0.03

0.59⫾0.02 0.56 0.72 0.66 0.65 0.76 0.69 0.70 0.76 0.69 0.69 0.63 0.52 0.78 0.58

0.08 0.05 0.06 0.05 0.03 0.05 0.05 0.04 0.05 0.05 0.07 0.14 0.03 0.04

0.04 0.05 0.04 0.04 0.03 0.06 0.03 0.04 0.04 0.04 0.04 0.04 0.07 0.04 0.24

a

Expt. QCBS EMS EMSa EMSbc ␣-H ␣-Ha ␣-Hbc ␤-H ␤-Ha ␤-Hbc CCC CCO CC␣ CO a

Experimental results from Ref. 7. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.144.81.13 On: Fri, 21 Nov 2014 07:48:51

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TABLE VI. Kinetic isotope effect, at 67.8 kcal/mol above the ZPE, obtained as k H /k D from trajectory calculations, RRKM theory, and experiment.

EMS EMSzpe ␣-X (X⫽H, D) ␤-X (X⫽H, D) CCO RRKMa RRKMb RRKMc Expt.d

(k H /k D) ␣ -D

(k H /k D) ␤ -D

0.92 1.09 0.92 0.79 1.00 1.15 1.11 1.00 2.00

0.95 1.15 1.05 0.94 1.22 1.19 1.11 1.00 1.09

a

With all the degrees of freedom treated as harmonic oscillators. With the two lowest frequency modes at the transition state treated as free rotors. c At a total energy of 140 kcal/mol and with the two lowest frequency modes at the transition state treated as free rotors. d At 65⫾10 kcal/mol. Taken from Ref. 4.


for the acetyl radical.1 These results may explain the different k H /k D ratios predicted for the CCO and ␤-X ensembles. As shown in a previous section, for the former ensemble there is a large percentage of rapid dissociations. This means that for these 共highly probable兲 events, only a reduced number of degrees of freedom are involved in the dissociation dynamics and probably the low-frequency modes are not included in the active modes. Thus, the above effects are not observed and a normal isotope effect is obtained. On the contrary, the initial excitation of the C␤ – H共D兲 local modes may involve the low frequency torsional modes in the vibrational energy relaxation process, leading to an inverse isotope effect.

b

3. Kinetic isotope effects

To analyze the influence of deuterium substitution on the rates of reaction, we have computed rate constants for CH3CH2CO, CD3CH2CO, and CH3CD2CO under several excitation schemes, namely EMS, EMSZPE, ␣ -X共X⫽H, D兲, ␤-X, and CCO. EMSZPE denotes an EMS ensemble of trajectories at 67.8 kcal/mol above the corresponding ZPE. Note that the ZPE varies upon isotopic substitution and so the total energies 共including the ZPE兲 for the above species differ under EMSZPE initialization. For the remaining ensembles, EMS, ␣-X, ␤-X, and CCO, a total energy of 112.0 kcal/mol 共including the ZPE兲 was employed. Table VI compares the k H /k D 共where k H and k D stand for the dissociation rates of the undeuterated and deuterated molecules, respectively兲 ratios calculated in this work with those obtained by Kim et al.4 As can be seen, our results cannot reproduce the experimental ratio of ⬃2 determined for CH3CD2CO. 4 The EMS and EMSZPE ensembles predict that the isotope effect for both deuterated species is not significant 共the ratios are close to unity兲. For ensembles ␤-X, the trajectory calculations predict a marked inverse isotope effect, that is, the rates for the deuterated species are larger than that for CH3CH2CO. Excitation of mode 571 共CCO ensemble兲, however, led to a normal isotope effect, similarly to that found for the acetyl radical.1 This result was explained on the basis of the different treatments for the lowfrequency vibrations as outlined below.1 For the sake of comparison and to provide further insights into the dynamics of the title reaction, we have also included the results obtained by the simple statistical RRKM theory. The RRKM ratios, which are slightly larger than unity, decrease when the lowest frequency modes at the transition state 共corresponding to the torsional modes兲 are treated as free rotors. This result comes from the fact that the free rotors provide a much larger sum-of-states contribution in the heavier molecules than in CH3CH2CO and thus the ratio k H /k D is lower with this treatment. On the other hand, our RRKM calculations predict that the ratio k H /k D would be reduced when the energy increases, as suggested previously

IV. CONCLUSIONS

Classical trajectories were performed to study the dissociation dynamics of the propionyl radical at energies ranging from 27.8 to 72.8 kcal/mol above the zero-point energy and under different types of initial conditions. An analytical potential function was first constructed from a PES developed previously for the acetyl radical,1 using additional data taken from the literature.7,8,12 Under microcanonical initial conditions, the dynamics was shown to be intrinsically RRKM for energies up to 42.8 kcal/mol, but intrinsically non-RRKM for higher energies. Several models based on the well-known RRK expression were used to explore the energy dependence of the microcanonical trajectory rates. The fits suggest that at the selected energies anharmonic effects, arising primarily from the two lowest frequency modes at the transition state, may play a significant role in the dynamics of the propionyl radical. The calculations predict that under selective excitations mode specificity may be exhibited in the propionyl radical. We concluded that the normal modes associated to the CCO fragment 共e.g., CCO bend or C␣ – H stretches兲 are strongly coupled to the reaction coordinate. On the other hand, the calculations involving the initial excitation of the C␤ – H local modes did not lead to enhancement of the dissociation rate. Therefore, our results corroborate the conclusions of Kim et al.4 indicated in the Introduction. The comparison of the product energy distributions calculated under EMS and QCBS initializations indicate that the ZPE leakage may be significant for trajectories initiated in the reactant phase space. For many of the ensembles studied, the average vibrational energy for the product CO is in good agreement with the experimental result. On the other hand, the comparison of the CO vibrational populations led us to conclude that one or more modes involving the motion of the CCO moiety may be significantly excited in the experiment of Hall et al.7 Finally, our calculations cannot predict the k H /k D ratio of ⬃2 observed for the deuterium substitution on the ␣-carbon.4 On the other hand, our results show an inverse isotope effect for the initial excitation of the ␤-H local modes and a normal isotope effect for the excitation of mode 571 共ensemble CCO兲. The comparison with the RRKM calculations led us to conclude again that, at the energies studies, the two lowest frequency modes at the transition state are more appropriately regarded as free rotors than as vibrations.



ACKNOWLEDGMENTS

A. P.-G. thanks XUNTA de Galicia for a grant. We also thank Centro de Supercomputation de Galica 共CESGA兲 for the use of their computational devices. 1

˜ a-Gallego, E. Martıńez-Nu´ñez, and S. A. Va´zquez, J. Chem. Phys. A. Pen 110, 11323 共1999兲. 2 ˜ ez and S. A. Va´zquez, Chem. Phys. Lett. 316, 471 E. Martıńez-Nun 共2000兲. 3 D. L. Bunker and W. L. Hase, J. Phys. Chem. 59, 4621 共1973兲; W. L. Hase, ibid. 90, 365 共1986兲. 4 S. K. Kim, J. Guo, J. S. Baskin, and A. H. Zewail, J. Phys. Chem. A 100, 9202 共1996兲. 5 J. C. Owrutsky and A. P. Baronavski, J. Chem. Phys. 108, 6652 共1998兲. 6 T. Shibata, H. Li, H. Katayanagi, and T. Suzuki, J. Phys. Chem. A 102, 3643 共1998兲. 7 G. E. Hall, H. W. Metzler, J. T. Muckerman, J. M. Preses, and R. E. Weston, Jr., J. Chem. Phys. 102, 6660 共1995兲. 8 E. Martıńez-Nu´ñez and S. A. Va´zquez, J. Mol. Struct. 共to be published兲. 9 E. Martıńez-Nu´ñez and S. A. Va´zquez, J. Chem. Phys. 107, 5393 共1997兲. 10 E. Martıńez-Nu´ñez and S. A. Va´zquez, J. Chem. Phys. 109, 8907 共1998兲. 11 E. Martıńez-Nu´ñez and S. A. Va´zquez, J. Phys. Chem. A 103, 9783 共1999兲. 12 K. W. Watkins and W. W. Thompson, Int. J. Chem. Kinet. 5, 791 共1973兲. 13 S. W. North, D. A. Blanck, J. D. Gezelter, C. A. Longfellow, and Y. T. Lee, J. Chem. Phys. 102, 4447 共1995兲. 14 See EPAPS Document No. E-JCPSA6-113-013046 for four pages including two tables and one figure. 共Table I: parameters for the analytical PES.


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Table II: vibrational populations for the product CO at 67.8 kcal/mol. Figure 1: lifetime distributions and single-exponential fits obtained for EMS initialization at the selected energies above ZPE.兲 The document may be retrieved via the EPAPS homepage 共http://www.aip.org/pubservs/ epaps.html兲 or from ftp.aip.org in the directory/epaps/. See the EPAPS homepage for more information. 15 D. L. Thompson, GENDYN program, Stillwater, 1991. 16 H. W. Schranz, S. Nordholm, and G. Nyman, J. Chem. Phys. 94, 1487 共1991兲; G. Nyman, S. Nordholm, and H. W. Schranz, ibid. 93, 6767 共1990兲. 17 C. Doubleday, Jr., K. Bolton, G. H. Peslberbe, and W. L. Hase, J. Am. Chem. Soc. 118, 9922 共1996兲. 18 W. L. Hase and D. G. Buckowski, Chem. Phys. Lett. 74, 284 共1980兲. 19 T. Baer and W. L., Hase Unimolecular Reaction Dynamics: Theory and Experiments 共Oxford University Press, New York, 1996兲. 20 R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombination Reactions 共Blackwell Scientific Publications, Oxford, 1990兲. 21 S. Chapman and D. L. Bunker, J. Chem. Phys. 62, 2890 共1975兲; C. S. Sloane and W. L. Hase, ibid. 66, 1532 共1977兲. 22 D. L. Bunker, Methods Comput. Phys. 10, 287 共1971兲, and references therein. 23 L. M. Raff and D. L. Thompson, in Theory of Chemical Reaction Dynamics, edited by M. Baer 共CRC, Boca Raton, 1985兲. 24 A. Einstein, Verh. Dtsch. Phys. Ges. 19, 82 共1917兲. 25 M. L. Brillouin, J. Phys. Radium 7, 353 共1926兲. 26 J. B. Keller, Ann. Phys. 共Paris兲 4, 180 共1958兲. 27 E. Martıńez-Nu´ñez and S. A. Va´zquez, J. Chem. Phys. 111, 10501 共1999兲. 28 K. Song and W. L. Hase, J. Chem. Phys. 110, 6198 共1999兲.
