Collision Frequency for Energy Transfer in Unimolecular Reactions

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Collision Frequency for Energy Transfer in Unimolecular Reactions Akira Matsugi J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b00444 • Publication Date (Web): 05 Feb 2018 Downloaded from http://pubs.acs.org on February 6, 2018

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The Journal of Physical Chemistry

Collision Frequency for Energy Transfer in Unimolecular Reactions Akira Matsugi* National Institute of Advanced Industrial Science and Technology (AIST), 16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan. *Corresponding author. E-mail: [email protected]

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Abstract Pressure dependence of unimolecular reaction rates is governed by the energy transfer in collisions of reactants with bath gas molecules. Pressure-dependent rate constants can be theoretically determined by solving master equations for unimolecular reactions. In general, master equation formulations describe energy transfer processes using a collision frequency and a probability distribution model of the energy transferred per collision. The present study proposes a novel method for determining the collision frequency from the results of classical trajectory calculations. Classical trajectories for collisions of several polyatomic molecules (ethane, methane, tetrafluoromethane, and cyclohexane) with monatomic colliders (Ar, Kr, and Xe) were calculated on potential energy surfaces described by the third-order density-functional tight-binding method in combination with simple pairwise interaction potentials. Low-order (including non-integer-order) moments of the energy transferred in deactivating collisions were extracted from the trajectories and compared with those derived using some probability distribution models. The comparison demonstrates the inadequacy of the conventional Lennard-Jones collision model for representing the collision frequency, and suggests a robust method for evaluating the collision frequency that is consistent with a given probability distribution model such as the exponential-down model. The resulting collision frequencies for the exponential-down model are substantially higher than the Lennard-Jones collision frequencies, and are close to the (hypothetical) capture rate constants for dispersion interactions. The practical adequacy of the exponential-down model is also briefly discussed.

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INTRODUCTION Collisional energy transfer plays a key role in unimolecular reaction kinetics in the limiting low-pressure and fall-off regions.1,2 Pressure-dependent rate constants for thermal unimolecular reactions can now be routinely calculated by solving unimolecular master equations3-5 using an empirical model adopted for the energy transfer process. At the same time, considerable effort has been devoted to transcending the empiricism by improving the knowledge of collisional energy transfer. A variety of experimental methods6-9 have been employed to directly or indirectly observe the collisional energy transfer rates of molecules. Classical trajectory calculations have also been used to study energy transfer.10-36 Early trajectory studies revealed many fundamental aspects of the energy transfer kernel such as the functional form of the energy transfer probability, the dependency of the energy transfer rate on bath temperature and type of third-body, and the effects of the initial vibrational and rotational energies. Some recent works have focused more specifically on the predictive ability of trajectory calculations; in particular, the studies reported by Jasper et al.31,34 have proposed that a master equation combined with a trajectory-based model of collisional transitions is able to predict pressure-dependent rate constants within typical experimental uncertainties. The present study is technically motivated by these preceding works. The present study deals with an issue that relates to collision frequencies (or total collision rate constants).21,37-42 This issue has been discussed several times in the literature, but seems to have long been postponed because the empirical approaches adopted so far to address the energy transfer process have obscured its physical importance. The energy transfer kernel appears in the two-dimensional master equation as R(E, J; Eʹ, Jʹ), which is the bimolecular rate coefficient for collisional transitions from the initial (pre-collision) energy and angular momentum (Eʹ, Jʹ) to the final (post-collision) energy and angular momentum (E, J). In many applications, the master equation is solved for a reduced one-dimensional form,1,43,44 in which 3



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the independent variable is the total energy (E) or the energy in the active degrees of freedom (ε). The rate coefficient for collisional energy transitions, R(E, Eʹ), is usually factorized into a collision frequency, Z(Eʹ), and a per-collision energy transfer probability function, P(E, Eʹ), as follows:

, ′ ≡ ′ , ′

(1)

This factorization, however, is artificial and made rather for convenience because Z(Eʹ) and P(E, Eʹ) cannot be chosen independently but must be consistent with one another. Nevertheless, almost all master equation calculations reported to date have taken Z(Eʹ) to be the energy-independent Lennard-Jones collision frequency, ZLJ, which is related to transport properties, in particular, viscosity. This collision frequency was merely introduced in the absence of any knowledge of appropriate alternatives,45 but has long served as a “standard” to facilitate intercomparison of energy transfer parameters derived for a variety of reaction systems. The Troe’s formulation of limiting low-pressure rate constants46,47 also employed ZLJ as a reference collision frequency. One rationalization41 of this practice is that the measurements of relative collision efficiency48-50 indicated that Z(Eʹ) is at least proportional to ZLJ for sufficiently large bath gases. Although any inadequacies of this reference can be compensated for via the total cross-section factor41,46 of the collision efficiency βc, this factor has been implicitly or explicitly assumed to be unity in many applications. Following the conventional practice has certain merits considering the extent of empirically accumulated knowledge of energy transfer parameters and collision efficiencies, whereas the development of physically more appropriate models is warranted to refine our understanding of the energy transfer process and make unimolecular reaction rate theory more predictive. Previous observations and indications related to the collision frequency may briefly be summarized as follows. Experimentally, anomalously large values of βc were inferred from 4


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measurements of the rate constant for the reaction H + O2 + M = HO2 + M with M = H2O.37,39 This anomaly was attributed to the inadequacy of ZLJ, and alternative models of the collision frequency were suggested. Hsu et al.37 used total quantum-mechanical scattering cross-sections,38 while Michael et al.39 employed a variant of a capture rate constant. Fernandes et al.40 also utilized capture rate constants for dispersion and dipole-dipole interactions in their analysis of low-pressure rate constants for the H + O2 + M reactions. These approaches could reduce βc to reasonable values, but their relevance to the energy transfer process remains uncertain. Michael et al. noted that their assignment of the collision frequency was based on a heuristic approach, which was adopted to reproduce the results of earlier classical trajectory calculations reported by Lendvay and Schatz21 and Brown and Miller.11 In these trajectory studies, different methods were used to determine collision frequencies. Lendvay and Schatz calculated the optimal maximum impact parameter, bmax, as

the minimum value needed to converge 〈∆〉 to within a threshold of 3%, where 〈∆〉

is the average energy transferred per collision. They found that the collision frequencies that corresponded to the optimal values of bmax were substantially larger than ZLJ by a factor of

1.3–4.7 (with an average of 3.1) for collisions of highly excited CS2, SF6, and SiF4 with a variety of bath gases. On the other hand, Brown and Miller directly calculated energy transfer cross-sections from trajectories for collisions of HO2 + He with sufficiently large bmax values. The cross-sections were binned with the amounts of energy transferred, and their distribution was fitted with an assumed functional form. The bin for the lowest amount of energy transferred was omitted from the fit to eliminate unwanted elastic collisions and make the cross-sectional function independent of bmax. The collision cross-section was then calculated as the integral of the derived function over the range of energy transferred. They found that the resulting cross-sections were consistently larger than the Lennard-Jones cross-sections. Hu and Hase14 employed similar approaches for the collisions of CH4 + Ar and reached the

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same conclusion. Michael et al. claimed that values obtained using such an approach were less secure than those of Lendvay and Schatz because the results were subject to imperfections in a fit.39 Nevertheless, it can also be said that the approach used by Brown and Miller is more consistent with the factorization expressed in eq 1. As indicated above, classical trajectory methods encounter inherent difficulty in defining a collision frequency, which is caused by the singularity that arises from a large number of elastic trajectories. This singularity certainly originates in classical mechanics and not occurs in quantum mechanics,11 but for highly excited polyatomic molecules, which are typically of interest in master equation calculations, the density of states can be significantly large that a similar singularity can be produced by a weak interaction with a collider. Also, because classical trajectories with finite bmax cannot be strictly elastic, hereafter they are described as “quasi-elastic”.29 The master equation is usually solved by employing a presumed functional form, the exponential-down model in most cases, for P(E, Eʹ). Therefore it is worthwhile in practice to establish a robust and consistent way to derive Z(Eʹ) that is appropriate for a given form of P(E, Eʹ). In the present study, a novel attempt is made to determine Z(Eʹ) in conjunction with P(E, Eʹ) based on the energy transfer moments. Low integer-order moments of the energy transferred have been considered as quantities that characterize the energy transfer kernel. Here, energy transfer moments are calculated from classical trajectories of the collisions of ethane (C2H6) with Ar, Kr, and Xe, methane (CH4) with Ar, tetrafluoromethane (CF4) with Ar, and cyclohexane (c-C6H12) with Ar. An inspection of the resultant moments, including those of non-integer-order, revealed an important aspect of their relevance to Z(Eʹ), which helped to propose a simple method to obtain plausible values of Z(Eʹ). The implications of the results for low-pressure rate constants are discussed through comparison of collision efficiencies derived from different collision models.

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COMPUTATIONAL METHODS Potential Energy Surface. Classical trajectory calculations were performed on potential energy surfaces that represent the intramolecular potentials of the selected target molecules (C2H6, CH4, CF4, or c-C6H12) and their interactions with the selected colliders (Ar, Kr, or Xe). The methods employed here are conceptually similar to those used in the recent studies of Jasper et al.30-33 The potential can be expressed as V = Vmolec + Vinter + Vcoll, where V, Vmolec and Vinter are the total, intramolecular and interaction potential energies, respectively, and Vcoll is the potential energy of the collider which is constant for the monatomic colliders studied in the present work. The intramolecular potentials were directly evaluated by the density-functional tight-binding (DFTB) method.51 The values of Vmolec and their gradients were calculated with the DFTB+ code52 using the third-order self-consistent-charge DFTB (DFTB3) method53 and the 3ob parameter set.54,55 The tolerance for the charge consistency was set at 10−10 to improve the energy convergence during the trajectory simulations. The vibrational frequencies of the target molecules computed by using the DFTB3 methods are compared with experimental values56 in Table 1. The root-mean-square deviation between the calculated and experimental values is 41 cm−1. The interaction potentials were approximated by sums of pairwise functions as

inter = M M

(2)

M = exp−# + %/ '

(3)

where i represents an atom in a target molecule, rM–i is the internuclear distance between

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collider M and atom i, and A, B, and C are the parameters for the Buckingham potential, VM–i. The parameters determined based on high-level focal-point analysis of the noble gas–CH4 and -CF4 interaction potentials57 were used.

Table 1. Computed and Experimental Vibrational Frequencies (cm−1) C2H6 lit.a

CH4 calc.b

lit.

CF4 calc.

d

lit. d

c-C6H12 calc.

c

lit. c

calc. c

289

274

1306

1296

435

407

248

211c

822c

851c

1534c

1474c

632d

577d

383

334

c

995

1015

2917

2859

909

876

426

444c

1190c

1228c

3019d

3052d

1281d

1143d

523

529

c

1379

1369

785

813c

1388

1394

802

803

1468c

1428c

863c

871c

1469c

1442c

907c

937c

2896

2874

1027c

1021c

2954

2900

1030

1049

c

2969

3011

c

1057

1067

2985c

3030c

1090

1094

1157

1155

1157

1202

c

1261

1278c

1266c

1280c

1347c

1380c

1355c

1382c

1383

1357

1437

1376

1437

1437

1443c

1441c

1457c

1448c

1465

1451

2852

2900

2860

2917

8


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a

Experimental values taken from ref 56.

b

Computed with DFTB3 method and 3ob parameter set.

c

Two-fold degeneracy.

d

Three-fold degeneracy.

2863c

2903c

2897c

2912c

2915

2988

2930

3006

c

2930

2988c

2933c

2996c

For comparison, trajectories of collisions of C2H6 with Ar were also calculated on the full-dimensional potential energy surface, which was directly evaluated by the second-order Møller-Plesset perturbation (MP2) theory using the 6-31+G(d,p) basis set with empirical scaling of correlation energies as described below. Figure 1 compares the potential energies determined by the MP2 method with those calculated by the explicitly correlated coupled-cluster method (CCSD(T)-F12)58-60 using the aug-cc-pVTZ basis set61,62 for 1000 geometries randomly selected from 400 trajectories of C2H6 + Ar collisions at 1000 K. These trajectories were calculated using the same method as described later but with impact parameters of zero and initial and final separations of 5.3 Å (10 bohr). The total (C2H6 + Ar) potential energies are plotted relative to those at an infinite separation with the equilibrium C2H6 structure. The interaction energy was calculated as the total potential energy relative to the sum of the potential energies of the isolated fragments with fixed intramolecular geometry. The CCSD(T)-F12/aug-cc-pVTZ energies are assumed to be highly accurate and serve as reference values. Figure 1 illustrates that the MP2/6-31+G(d,p) level of theory systematically overestimate

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both the total and interaction energies. The deviations result from basis set incompleteness and superposition errors as well as higher-order correlations. These errors could be compensated for by artificially scaling the spin components of the correlation energy; the correlation energy of the spin-component-scaled (SCS)-MP2 method63 is given as

( = )* * + )+ +

where *

and +

(4)

are the antiparallel- and parallel-spin pair contributions to the MP2

correlation energy, respectively, and pS and pT are scaling factors. These factors are associated with dynamical (pS) and nondynamical (pT) correlation effects, the latter of which is usually overestimated by the MP2 method. The original SCS-MP2 method was developed to correct for the correlation energy, and the scaling factors are chosen so that the scaled total correlation energy roughly equals that estimated by the MP2 method. In the present study, however, no such constraint was assumed, because the present scaling is also intended to account for deficiencies in the basis set. By employing values of pS = 1.8 and pT = 1.0, the SCSʹ-MP2/6-31+G(d,p) method (the prime symbol denotes the use of nonstandard scaling factors) provided a significant improvement in agreement with the reference values of both the total and interaction energies with no systematic deviations, as shown in Fig. 1. The root-mean-square deviations from the reference energy values are 4.8 and 0.57 kJ mol−1 for the total and interaction energies, respectively. These scaling parameters were used in the direct trajectory calculations. The (SCSʹ-)MP2 and CCSD(T)-F12 calculations were performed using the Gaussian 0964 and Molpro 2012.165 programs, respectively.

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Figure 1. Comparisons of the relative total potential energies (top) and the interaction energies (bottom) calculated at the MP2/6-31+G(d,p) (crosses) and SCSʹ-MP2/6-31+G(d,p) (circles) levels of theory to those calcualated at the CCSD(T)-F12/aug-cc-pVTZ level for 1000 geometries selected from the trajectories of C2H6 + Ar collisions.

Trajectory Simulations. The initial conditions for classical trajectories were generated by using a scheme similar to that employed previously.36,66 Firstly, the initial vibrational energy, Evib, of a molecule was distributed randomly among the normal modes,67 assuming that the system can be described as a collection of separable harmonic oscillators. Specifically, the vibrational energies associated to each mode i (Ei) were successively determined using the following expression

11



1

1

= ,-./ − 0 3 41 − 6789 : 021

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(5)

where Ri represent random numbers ranging uniformly between 0 and 1, and nvib is the number of vibrational modes. The energies of each mode, Ei, were then augmented with the individual zero-point energies, and transformed into the normal-mode coordinates and momenta with randomly determined vibrational phases.67 The normal mode displacements and momenta were iteratively scaled so that the total vibrational energy matched the desired value to within 1 cm−1. At high Evib, the phase space sampled by this procedure significantly deviates from that of coupled anharmonic oscillators. Therefore, trajectories of isolated molecules were run for 1000 fs starting from the generated coordinates and momenta to facilitate the redistribution of vibrational energy. The resultant coordinates and momenta constituted the initial vibrational state and phase of the molecules. The initial rotational state was then generated using quasi-classical rigid-rotor sampling.67 The principal axes and associated moments of inertia were calculated for the instantaneous coordinates generated by the abovementioned method. The rotational quantum numbers J and K were generated from the thermal distribution specified by the initial rotational temperature Trot, and the components jx, jy, and jz of the angular momentum j (= ;? = @ℏ

(6a)

>A = B> − >? sin2E

(6b)

>F = B> − >? cos2E

(6c)

where R is a random number. Then, the molecule was randomly rotated through its Euler angles to give a random orientation, and placed together with the collider at a center-of-mass separation x0 with an impact parameter b. To improve the convergence, the impact parameter 12


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was uniformly sampled over the range of 0–bmax, instead of sampling b2 uniformly over the range of 0–bmax2.68 The relative velocity of the molecule and collider was determined from the relative translational energy, Erel, which was sampled from the probability distribution

)IJK dIJK =

IJK IJK exp P− Q dIJK M N O MN O

(7)

where kB is the Boltzmann constant and T is the bath gas temperature. The trajectories were integrated using the velocity Verlet algorithm with a time step of 0.08 fs. The initial and final separations were set at x0 = 10.6 Å (20 bohr) and the maximum impact parameter was bmax = 9.5 Å (18 bohr) except for the collisions of c-C6H12 with Ar, for which x0 = 15.9 Å (30 bohr) and bmax = 13.8 Å (26 bohr) were employed. On the DFTB3 with the pairwise interaction potentials, batches of 4000 trajectories were run under each set of conditions specified by Evib and T, and for each combination of target molecule and collider: C2H6 + Ar/Kr/Xe, CH4 + Ar, CF4 + Ar, and c-C6H12 + Ar. The calculations were carried out mainly for Evib = 400 kJ mol–1 and T = 300, 1000, 2000, and 3000 K with Trot = T. This Evib was selected so as to have a value close to typical C-H and C-C bond dissociation energies. Results with Evib = 200 kJ mol–1 are also presented for the C2H6 + Ar collisions. For the SCSʹ-MP2 potential, 600 trajectories were calculated for the C2H6 + Ar collisions at each of the temperatures T = 1000 and 3000 K with Evib = 400 kJ mol–1. The total energy transferred per collision was calculated as the change in the internal energy of the target molecules. All error limits reported for the parameters derived from the trajectory calculation correspond to two standard errors throughout the paper.

TRAJECTORY RESULTS AND ENERGY TRANSFER MODELS Table 2 lists some downward energy transfer moments obtained from the trajectory 13



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calculations as

〈∆d 〉RI S 6

VW

1 2 = ′ − 6 TU

(8)

21

where bi, Eiʹ, and Ei are the impact parameter and the initial and final rovibrational energies of the molecule, respectively, for the i-th deactivating trajectory, and Nd is the number of trajectories that resulted in deactivation (Eiʹ > Ei). The weighting factor, 2bi/bmax, was added to correct for bias in the selection of the impact parameter. Here, only downward transfer moments are of concern because upward transfer rates can be enforced by the detailed balance in typical master equation applications. Since the moments are defined as eq 8 in this study, the order, n, can have any non-integer values. The moments listed correspond to a given value of bmax and can be scaled11,30-32 according to an assumed collision frequency, Z, as

〈∆d 6 〉 =

traj 〈∆d 6 〉RI S

(9)

where Ztraj is the hard-sphere collision frequency for a collision radius bmax:

traj = E ;8MN O/E[

(10)

where µ is the reduced mass of two colliding molecules. This scaling method leaves the

products 〈∆d 6 〉 unchanged and effectively eliminates (quasi-)elastic collisions given that Z is sufficiently large and an appropriate function P(E, Eʹ) is chosen.

At first, following the convention for master equation calculations, the Lennard-Jones collision frequency

LJ = E^LJ ;8MN O/E[ _ ,∗

(11)

14


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is employed to scale the moments, where Ω(2,2)* is the reduced collision integral, which can be approximately given as47 1

_,∗ ≈ b0.636 + 0.567 log1k MN O/ lLJ m

(12)

and σLJ and εLJ are the parameters of the Lennard-Jones potential. These parameters were estimated from the pure gas values by combining rules (arithmetic mean for σLJ and geometric mean for εLJ). The following pure gas values were used: (σ/Å, (ε/kB)/K) = (4.44, 216), (3.76, 149), (4.66, 134), (6.18, 297), (3.54, 93), (4.05, 231) and (3.66, 179) for C2H6, CH4, CF4, c-C6H12, Ar, Kr, and Xe, respectively.69 The most common form of P(E, Eʹ) is the exponential-down model,1,2,46 which assumes the transition probability distribution of

, ′ ∝ exp 4−

′ − : o

(13)

for deactivating collisions (Eʹ > E) and the probabilities for activating transitions are determined by detailed balance. The parameter α is equivalent to the average energy

transferred in deactivating collisions, 〈∆d 〉, at sufficiently high energies. Values of α that are

consistent with ZLJ can be calculated as opq = RI S /pq 〈∆d 〉RI S , which are listed in Table

2. The values of αLJ increase with temperature, and if they are assumed to be proportional to some power of the temperature, the temperature exponents were found to be in the range of 0.63–0.81. The dependence of the results on a given value of Evib was examined for the C2H6 + Ar collisions. As expected based on previous studies,36,66 the energy transfer moments were found to be insensitive to the initial vibrational energies. The values of the parameters αLJ obtained from the trajectories on potential energy surfaces at the two levels of theory (DFTB3 with the pairwise interaction and SCSʹ-MP2) agree with each other within the limits of statistical uncertainty. The agreement justifies the use of the DFTB3 intramolecular and the parameterized interaction potentials for the characterization of collisional energy transfer

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processes. Unless otherwise noted, the results presented hereafter refer to those for the trajectories on the DFTB3 with the pairwise interaction potentials with Evib = 400 kJ mol−1.

Table 2. Calculated Downward Energy Transfer Moments and Energy Transfer Parameters for the Lennard-Jones Collision Model system

T/K

〈∆d 〉 /

Ztraj / 10−10

cm3

〈∆d1/ 〉 〈∆d 〉1/

cm−1

/ cm−1

/ cm−1

molecule−1

C2H6 + Ar

C2H6 + Arb C2H6 + Kr

C2H6 + Xe

CH4 + Ar

cm3 −1 −1

s

a

αLJ / cm−1

molecule

−1

C2H6 + Ar

ZLJ / 10−10

s

300

17.3

47.1 ± 3.9

17.3 ± 1.6

125 ± 11

3.71

220 ± 18

1000

31.7

71.8 ± 6.1

19.4 ± 1.8

225 ± 17

4.98

456 ± 39

2000

44.8

108 ± 10

25.5 ± 2.6

367 ± 37

6.11

793 ± 75

3000

54.8

140 ± 14

30.0 ± 3.2

511 ± 53

6.94

1102 ± 107

300

17.3

43.4 ± 3.5

16.4 ± 1.5

108 ± 9

3.71

203 ± 16

1000

31.7

77.1 ± 6.9

21.7 ± 2.1

242 ± 25

4.98

490 ± 44

2000

44.8

101 ± 9

23.4 ± 2.5

337 ± 30

6.11

739 ± 68

3000

54.8

147 ± 14

30.7 ± 3.3

532 ± 55

6.94

1165 ± 112

1000

31.7

75.0 ± 16.0

25.7 ± 5.4

241 ± 49

4.98

477 ± 101

3000

54.8

148 ± 37

36.4 ± 9.4

532 ± 131

6.94

1169 ± 297

300

15.3

50.1 ± 3.6

21.0 ± 1.7

115 ± 8

3.73

205 ± 15

1000

27.9

85.2 ± 7.5

25.5 ± 2.4

255 ± 23

4.86

489 ± 43

2000

39.4

116 ± 11

30.2 ± 3.1

361 ± 29

5.90

776 ± 71

3000

48.3

158 ± 15

38.0 ± 4.2

513 ± 53

6.67

1145 ± 105

300

14.5

59.6 ± 4.3

27.8 ± 2.2

127 ± 10

4.07

213 ± 15

1000

26.5

85.4 ± 7.1

28.3 ± 2.6

235 ± 19

5.24

432 ± 36

2000

37.5

133 ± 11

39.0 ± 3.8

383 ± 31

6.33

786 ± 65

3000

45.9

173 ± 16

44.4 ± 4.6

535 ± 50

7.15

1112 ± 100

300

21.2

26.8 ± 2.1

9.7 ± 0.8

73 ± 6

3.60

158 ± 12

1000

38.8

37.5 ± 3.4

9.9 ± 0.8

131 ± 15

4.90

297 ± 27

2000

54.8

56.5 ± 6.5

11.6 ± 1.1

244 ± 37

6.04

513 ± 59

3000

67.1

67.6 ± 6.5

13.6 ± 1.3

276 ± 29

6.88

660 ± 63

16


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CF4 + Ar

c-C6H12 + Ar

300

13.7

61.0 ± 5.3

22.8 ± 2.0

167 ± 20

2.89

289 ± 25

1000

25.0

119 ± 10

31.5 ± 3.1

363 ± 28

3.95

754 ± 62

2000

35.4

196 ± 18

43.7 ± 4.6

671 ± 67

4.87

1422 ± 131

3000

43.3

231 ± 22

47.8 ± 5.3

806 ± 66

5.56

1803 ± 168

300

28.8

52.5 ± 5.2

14.4 ± 1.5

163 ± 19

4.60

328 ± 33

1000

52.6

66.5 ± 6.3

13.9 ± 1.5

233 ± 20

6.09

574 ± 55

2000

74.4

99.6 ± 10.8

16.2 ± 2.0

402 ± 39

7.44

996 ± 108

3000

91.1

130 ± 14

21.0 ± 2.5

550 ± 75

8.44

1405 ± 154

a

−1

Evib = 200 kJ mol .

b

On the SCSʹ-MP2 potential energy surface.

In the following, the suitability of the models of collision and energy transfer is assessed by inspecting the downward energy transfer moments. The population moments for downward energy transfer are given by st

〈∆d 〉 = r d ′ − 6 , ′ 6

(14)

k

In practice, the lower limit of the integration can be replaced with −∞,46 and the moments for the exponential-down model can be expressed as

〈∆d 6 〉 = Γv + 1o 6

(15)

where Γ is the Gamma function. These may be related to trajectory-based moments as

defined by eq 9, and a comparison can be made by rescaling 〈∆d 6 〉 so as to be consistent

with Ztraj. Figure 2 compares the trajectory results, 〈∆d 6 〉RI S 1/6

values from the model, i.e., bpq /RI S 〈∆d 6 〉m

1/6

, with the corresponding

, which are calculated using eq 15 with α

= αLJ, for the C2H6 + Ar collisions at T = 1000 K. As can be seen, the dependence on the value of n of the moments is not properly accounted for by the exponential-down model with Z = ZLJ. Essentially similar results were obtained for the different molecules, colliders, and 17



temperatures investigated in the present study. The trajectory results for a maximum impact parameter reduced to 70% of its original value are also plotted in Fig. 2. These moments are shown scaled by a factor of 0.491/n to be compared with the results for the original value of bmax. Since this reduction in the value of bmax only reduces the number of quasi-elastic collisions, it does not have a discernible effect on their n-dependence at the order close to or greater than 1. Therefore, the discrepancy should be attributed to inadequacy of the model employed.

Figure 2. Comparison of trajectory-based downward energy transfer moments (circles) with those obtained using the exponential-down model with Z = ZLJ (solid line; scaled to be consistent with Ztraj (see text)) for collisions of C2H6 with Ar at T = 1000 K. The trajectory results for bmax reduced to 70% of its original value are also shown (crosses).

Similar comparisons were carried out using different functional forms of P(E, Eʹ) for deactivating collisions, namely, the exponential power distribution model

18


Page 18 of 57

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, ′ ∝ exp w− 4 〈∆d 6 〉 =

′ − : x oF F

(16)

Γv + 1/y 6 oF Γ1/y

(17)

and the double exponential distribution model

, ′ ∝ 1 − z exp 4−

′ − ′ − : + z exp 4− : o(1 o(

〈∆d 6 〉 = Γv + 1{1 − z o(1 6 + zo( 6 |

(18)

(19)

where y, αy, c, αc1, and αc2 are parameters that describe the shapes of the distributions as well as the energy transfer moments. As in the case of the exponential-down model mentioned

above, the first-order moment, 〈∆d 〉RI S , scaled by Ztraj/ZLJ is used to constrain the parameters; for example, the parameter αy that is consistent with ZLJ for a given value of y is given as

oF,pq =

Γ1/y traj Γ1/y 〈∆d 〉RI S = o Γ2/y pq Γ2/y pq

(20)

The top panel of Fig. 3 shows the rescaled downward energy transfer moments, bpq / 1/6

RI S 〈∆d 6 〉m

, which were calculated using eq 17 with αy = αy,LJ at several values of y.

The shape of this distribution is governed by the parametric exponent y. If y = 1, this model is equivalent to the exponential-down model, eq. 13. As y decreases, the tail of the distribution is extended and the values of the high-order moments increase. On the other hand, a higher value of y makes the distribution more compact, which results in a decrease in the values of high-order moments. Figure 3 shows that the exponential power distribution model with y = 2–4 provides a better representation of the moments for n > 1 than the exponential-down model. However, the n-dependence of the moments near n = 1 is still not properly described; the energy transfer moments obtained from the model have the steeper dependence on the 19



value of n than the trajectory results. Figure 3 also shows the results for the double exponential distribution model. Since three parameters are required, results are presented for the following specific cases: c = 0.1 and αc2/αc1 = 10, c = 0.2 and αc2/αc1 = 5, and c = 0.5 and

αc2/αc1 = 2. These models necessarily give wider distributions than the single exponential-down model and, hence, always result in a steeper n-dependence of the energy transfer moments.

Figure 3. Comparison of the trajectory-based downward energy transfer moments (circles) with those obtained using the exponential power (top) and double exponential (bottom) distribution models with Z = ZLJ (lines; scaled to Ztraj) for collisions of C2H6 with Ar at T = 1000 K.

20


Page 20 of 57

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The results described above suggest that the distribution models considered here cannot adequately represent the energy transfer process if ZLJ is used as the collision frequency. As stated in the Introduction section, Z should be consistent with P(E, Eʹ) and there is no solid reason to employ ZLJ. Given that the energy transfer process can be characterized by low-order energy transfer moments, Z should be derived from these moments in conjunction with the parameters used for P(E, Eʹ). The principal difficulty encountered with Z = ZLJ is that the n-dependence of 〈∆d 6 〉RI S

1/6

near n = 1 cannot be reproduced using the models of P(E,

Eʹ) employed. Therefore, a straightforward method of dealing with this difficulty comprises the direct calculation of Z from the energy transfer moment and its derivative with respect to n. For simplicity, results will only be given for the exponential power distribution (eq 16) model, but the method proposed below can be applied to any functional form of P(E, Eʹ). For the exponential power distribution model with a given value of y, the corresponding collision frequency Zy as well as the parameter αy are related through eqs 9 and 17 to the trajectory-based moment and its derivative with respect to n at n = 1 as

traj Γ2/y 〈∆d 〉RI S oF = Γ1/y F

(21)

traj Γ2/y }2/y 〈∆d ln ∆d 〉RI S oF wln oF + x= Γ1/y y F

(22)

and

respectively, where ψ is the digamma function, ψ(x) = Γʹ(x)/Γ(x), and 〈∆d ln ∆d 〉RI S is given by

21



〈∆d ln ∆d 〉RI S

VW

1 2 = ′ − ln ′ − TU 21

Page 22 of 57

(23)

Therefore, the parameter and the collision frequency can be obtained as

ln oF =

〈∆d ln ∆d 〉RI S }2/y − 〈∆d 〉RI S y

F = traj

(24)

Γ1/y 〈∆d 〉RI S Γ2/y oF

(25)

The results for y = 1 (equivalent to the exponential-down model; the parameters are denoted as α1 and Z1) and y = 0.5 (α0.5 and Z0.5) are listed in Table 3. The values for different y can be calculated from the dependences of αy and Zy on the values of y as shown in Fig. 4. The collision frequencies Zy depend on the value of y and are subsequently higher than ZLJ, by factors of 1.4–1.9 for y = 1 and 1.8–2.6 for y = 0.5, respectively. Accordingly, the average downward energy transfer per collision, 〈∆d 〉, is less than that obtained in cases of Z = ZLJ,

i.e., αLJ. The results for the reduced value of Evib listed in Table 3 suggest that Zy is independent of the initial vibrational energy, which rationalizes the use of an energy-independent value of the collision frequency. A comparison of values of 〈∆d 6 〉RI S

1/6

1/6

present model, i.e., bF /RI S 〈∆d 6 〉m

with the corresponding values derived using the

, is shown in Fig. 5 for the C2H6 + Ar collisions at

T = 1000 K. The use of Zy as the collision frequency clearly provides an improved description of the energy transfer moments, in particular for y = 0.5. Figure 6 shows the results for all systems that are presently studied. In all cases, the exponential power distribution model with Z = Zy and y = 0.5 gave results that closely corresponded to the trajectory results. There are still some deviations in the region with low n values (i.e., n < 0.5), which originate from quasi-elastic collisions (and numerical errors, as will be seen later) at large b values. Indeed,

22


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the modeled values are similar to the trajectory results for the reduced bmax as seen in Fig. 5. The potential contribution of quasi-elastic collisions to the overall energy transfer process and unimolecular reaction kinetics is discussed in the next section.

Table 3. Collision Frequencies and Energy Transfer Parameters Calculated Using Equations 24 and 25 system

T/K

Z1 / 10−10 cm3

α1 / cm−1

molecule−1 s−1 C2H6 + Ar

a

C2H6 + Ar

b

C2H6 + Ar

C2H6 + Kr

C2H6 + Xe

CH4 + Ar

Z0.5 / 10−10 cm3

α0.5 / cm−1

molecule−1 s−1

300

6.02 ± 0.40

136 ± 14

8.11 ± 0.54

16.8 ± 1.7

1000

7.57 ± 0.48

301 ± 26

10.2 ± 0.6

37.2 ± 3.2

2000

9.33 ± 0.70

519 ± 56

12.6 ± 0.9

64.2 ± 7.0

3000

10.1 ± 0.8

757 ± 83

13.6 ± 1.1

93.7 ± 10.3

300

6.48 ± 0.40

116 ± 10

8.73 ± 0.54

14.4 ± 1.3

1000

8.13 ± 0.61

300 ± 32

10.9 ± 0.8

37.2 ± 3.9

2000

9.38 ± 0.66

482 ± 47

12.6 ± 0.9

59.6 ± 5.8

3000

10.1 ± 0.8

804 ± 87

13.5 ± 1.1

99.5 ± 10.8

1000

8.62 ± 1.52

276 ± 76

11.6 ± 2.05

34.1 ± 9.4

3000

11.1 ± 2.1

734 ± 222

14.9 ± 2.84

90.9 ± 27.5

300

6.55 ± 0.36

117 ± 9

8.82 ± 0.48

14.5 ± 1.1

1000

7.56 ± 0.49

314 ± 31

10.2 ± 0.7

38.9 ± 3.8

2000

9.34 ± 0.61

491 ± 45

12.6 ± 0.8

60.7 ± 5.6

3000

10.8 ± 0.8

706 ± 71

14.6 ± 1.1

87.4 ± 8.8

300

7.15 ± 0.37

121 ± 9

9.62 ± 0.50

15.0 ± 1.2

1000

8.25 ± 0.49

274 ± 25

11.1 ± 0.7

34.0 ± 3.1

2000

10.4 ± 0.7

478 ± 42

14.0 ± 0.9

59.1 ± 5.2

3000

11.1 ± 0.8

716 ± 69

14.9 ± 1.0

88.7 ± 8.5

300

6.83 ± 0.39

83 ± 7

9.20 ± 0.53

10.3 ± 0.9

1000

8.20 ± 0.64

177 ± 21

11.0 ± 0.9

22.0 ± 2.6

2000

8.23 ± 0.80

376 ± 60

11.1 ± 1.1

46.6 ± 7.5

23



CF4 + Ar

c-C6H12 + Ar

Page 24 of 57

3000

10.5 ± 0.9

431 ± 49

14.2 ± 1.2

53.4 ± 6.1

300

4.71 ± 0.36

177 ± 21

6.35 ± 0.48

22.0 ± 2.6

1000

6.13 ± 0.39

486 ± 40

8.25 ± 0.53

60.1 ± 4.9

2000

7.20 ± 0.54

963 ± 102

9.69 ± 0.72

119 ± 13

3000

8.21 ± 0.58

1220 ± 114

11.1 ± 0.8

151 ± 14

300

7.46 ± 0.57

203 ± 25

10.1 ± 0.8

25.1 ± 3.1

1000

9.92 ± 0.71

353 ± 34

13.4 ± 1.0

43.6 ± 4.2

2000

10.5 ± 0.9

705 ± 75

14.1 ± 1.2

87.3 ± 9.3

3000

12.8 ± 1.2

924 ± 118

17.3 ± 1.6

114 ± 15

a

Evib = 200 kJ mol−1.

b

On the SCSʹ-MP2 potential energy surface.

Figure 4. Dependences of Zy and αy on the value of y.

24


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Figure 5. Comparison of trajectory-based downward energy transfer moments (circles) with those obtained using the exponential power distribution model with Z = Zy (lines; scaled to Ztraj) for collisions of C2H6 with Ar at T = 1000 K. The trajectory results for the reduced value of bmax are also shown (crosses).

Figure 6. Comparison of trajectory-based downward energy transfer moments (symbols)

25



Page 26 of 57

with those obtained using the exponential power distribution model with Z = Zy and y = 1 (solid lines) and y = 0.5 (dashed lines). The modeled moments are scaled so as to be consistent with Ztraj.

DISCUSSION Collision Efficiency. The effects of the selection of different Z values on the collision efficiency, βc, are examined here, using the case of C2H6 + Ar at T = 1000 K as an example. The collision efficiency can be defined as the ratio of the limiting low-pressure rate constant, k0, to the hypothetical strong collision rate constant, Mk*~ , as46

( ≡ Mk /Mk*~

(26)

where

Mk*~ ≡ {M| pq r d

(27)

s

where [M] denotes the concentration of the bath gas M, E0 is the threshold energy for dissociation (367 kJ mol−1 for C2H647), and f(E) is the Boltzmann equilibrium distribution. The limiting low-pressure rate constant k0 is derived as a solution to the master equation; for a one-dimensional case with an energy-independent collision frequency, k0 can be expressed as s

Mk = {M| r d′ r d , ′′ k

s

26


(28)

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where g(E) is the steady-state distribution. In the Troe’s formulation,46 ZLJ is employed as a

reference collision frequency in calculations of Mk*~ , and βc is factorized into the components,

βcσ and βc∆Ε, of which the former accounts for deviation from the reference ZLJ and the latter

is expressed as follows:

(∆s =

k d′ s d , ′′

s

s d

(29)

To derive βc∆E, the steady-state distribution is obtained by solving the master equation numerically. The exponential power distribution model (eq 16) is assumed for deactivating collisions, and the activation probability is determined by detailed balance. The density of states of C2H6 was calculated by the modified Beyer−Swinehart algorithm1 using experimental values of vibrational frequencies56 (as listed in Table 1) and rotational constants70,71 (A = 2.671 cm−1 and B = 0.6631 cm−1). An energy grains size of 1 cm−1 was employed to ensure convergence of the results at low values of y. The states involve all the rotational and vibrational degrees of freedom, which is in accordance with the trajectory results that relate to the transfer of the total rovibrational energy. At high temperatures, this treatment was shown to give values of low-pressure rate constants similar to those determined using an approximated two-dimensional model.44 The results of numerical calculations of βc∆Ε for Z = ZLJ and Z = Zy are presented in the top panel of Fig. 7, which shows βc∆Ε as a function of y. If a constant value of ZLJ is employed as the collision frequency, βc∆Ε ≈ 0.1 and is insensitive to the value of y. For the model with Z = Zy, the values of βc∆Ε are approximately half of those with Z = ZLJ and exhibit a positive dependence on the value of y. This dependence is partially due to the dependence of Zy on the value of y (Fig. 4). As the limiting low-pressure rate constant depends on the product of Z and

βc∆Ε, a similar comparison made for Zβc∆Ε is shown in the bottom panel. It can be seen that

27



Page 28 of 57

the dependence of βc∆Ε on the value of y is compensated for by the corresponding dependence of Zy, and the product is nearly independent of y; the value of Zyβc∆Ε decreases by less than 5% from y = 1.0 to 0.5. This independence indicates that Zyβc∆Ε is insensitive to high-order energy transfer moments, which are comparatively sensitive to the value of y as seen in Figs. 5 and 6. Therefore, although the trajectory results suggest y ~ 0.5, the exponential-down model with Z = Z1 can be a practical alternative in master equation applications, at least for single-channel reactions. It should be noted, however, that the same may not be applicable to multichannel unimolecular reactions which have multiple dissociation channels with different threshold energies.

The symbols plotted in Fig. 7 represent the approximate analytical solution for the exponential-down model46:

(∆s o ≈ P

o Q o + s MN O

(30)

where FE is the energy dependence factor of the density of states. As shown, the analytical solutions are virtually identical to the numerical ones at y = 1. Eq. 30 provides a simple estimate of the consequence of employing different values of Z in the derivation of Zβc∆Ε from the trajectory results. Given the relation α1Z1 = αLJZLJ, the ratio of Zβc∆Ε for Z = Z1 to that for Z = ZLJ can be given as

1 + opq /s MN O 1 (∆s o1 ≈ 1 pq 4 : pq (∆s opq 1 + pq opq /s MN O

(31)

which converges to ZLJ/Z1 and Z1/ZLJ at the diffusion limit (opq /s MN O ≪ 1) and the strong

collision limit (opq /s MN O ≫ 1), respectively.

28


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Figure 7. Comparisons of βc∆Ε and Zβc∆Ε calculated using the exponential power distribution model with Z = ZLJ (dashed lines) and Z = Zy (solid lines) as a function of y for collisions of C2H6 with Ar at T = 1000 K.

Quasi-Elastic Collisions. A detailed characterization of quasi-elastic collisions is not a concern of the present study, because it would be hampered by the use of classical mechanics as well as by numerical integration errors. Instead, the potential effects of quasi-elastic collisions on low-pressure rate constants are assessed. Quasi-elastic collisions appear as a singularity in the probability distribution of the energy transferred near E = Eʹ, and primarily arise from trajectories with large impact parameters. They hardly contribute to (the properly scaled) energy transfer moments at the values of n

29



Page 30 of 57

close to or greater than 1, but the moments at values of n close to 0 can be significantly affected. This is shown in Fig. 5 as the difference in the value of 〈∆d 6 〉RI S

1/6

calculated

from the trajectories with the original and reduced bmax. The situation is illustrated in Fig. 8 in which the absolute values of the energy transferred are plotted as a function of b/bmax. The average energy transferred in collision is ~300 cm−1 at small b. The amount of energy transferred decreases sharply (note the logarithmic scale on the axis) at values of b greater than ~0.4bmax, and only less than a few cm–1 of the energy can be transferred at b > 0.7bmax. In addition, the decrease in the energy transferred becomes gentle at values of b greater than ~0.8bmax. This is because numerical errors in the trajectory integration and gradient evaluation exceed the amount of energy transferred at large b. Therefore, the deviation seen in Fig. 5 in the region with low n values is caused, at least in part, by the numerical errors as well as quasi-elastic (classical) collisions. In any case, Fig. 8 shows that only insignificant amounts of energy (less than 1 cm−1 in most cases) can be transferred at large b. Here, an artificial model that includes a term mimicking quasi-elastic collisions, namely,

, t ∝

1 t − exp w− 4 : x + t − | ∆ , TF oF F

(32)

is considered, where Ny is a normalizing constant for the first term, TF = oF Γy + 1/y, fQE is a parameter that corresponds to the relative number of quasi-elastic collisions, U is a uniform distribution function describing the quasi-elastic collisions, which is defined as

1 for ∈ − , + 2 2 | , =

0 otherwise

(33)

and ∆EQE and δEQE represent the center and width, respectively, of the distribution of the energy to be transferred in quasi-elastic deactivating collisions. For pragmatic reasons ∆EQE and δEQE are assumed to have a constant value that is equal to the energy grain size of 1 30


Page 31 of 57 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


cm−1; the quasi-elastic collisions are assumed to transfer energies to the neighboring grains. The collision frequency of this model is Z = Zy + fQEZy, where the second term represents quasi-elastic collisions. Figure 9 shows a comparison of the scaled moments derived using the model with fQE = 0, 2, and 5. The trajectory results can be approximately reproduced by setting fQE = 2. The model with fQE = 5 represents a more extreme case, in which the artificial quasi-elastic collisions constitute a large proportion of total collisions. The collision efficiency factor βc∆Ε was numerically calculated for each case. The product, Zβc∆Ε, was found to depend only slightly on the value of fQE, and had values of Zβc∆Ε = 3.99 × 10−11, 3.76 × 10−11, and 3.71 × 10−11 cm3 molecule−1 s−1 at fQE = 0, 2, and 5, respectively. Even though the value of ∆EQE was arbitrarily selected, the energy transferred in most collisions at large b was less than 1 cm−1, and the results presented here reveal the insignificance of quasi-elastic collisions for unimolecular reaction kinetics.

Figure 8. Absolute values of energy transferred in collisions of C2H6 with Ar at T = 1000 K as a function of b/bmax. The line drawn approximately traces the average.

31



Figure 9. Comparison of trajectory-based downward energy transfer moments (circles) with those derived using the artificial model represented by eq 32 with Z = Zy + fQEZy and y = 0.5 (lines; scaled to Ztraj) for collisions of C2H6 with Ar at T = 1000 K. The trajectory results for the reduced values of bmax are also shown (crosses).

Collision Frequency. As demonstrated above, collision frequencies that are appropriate for modeling collisional energy transfer depend on an assumed functional form of P(E, Eʹ), and are subsequently higher than ZLJ for the functions considered in the present study. The conventional exponential-down model is shown to be reasonable for describing unimolecular reaction kinetics at low pressures if the collision frequency is chosen appropriately. Therefore, it would be valuable for future master equation studies to have a convenient expression for Z that can represent Z1. The trajectory results suggest that the collision frequency is insensitive to the vibrational energy, which indicates that it can be described based only on dynamics of intermolecular

32


Page 32 of 57

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degrees of freedom but not on intramolecular ones. A consequence of the intermolecular interaction in a collision between rigid molecules appears in the form of the scattering angle; therefore, potential relationships between the collision frequency and the scattering angle are investigated here with the intention of finding a simplified representation of Z1 in a heuristic manner. For the sake of simplicity, the investigation is made for ensembles of trajectories starting with fixed relative translational energies of Erel = 5 and 20 kJ mol−1 for collisions of C2H6 with Ar at Evib = 400 kJ mol−1 and Trot = 1000 K. Batches of 4000 trajectories were run for each value of Erel and the impact parameter was uniformly sampled over the range of 0– bmax, where bmax = 9.5 Å (18 bohr). The scattering angles (the angles of deflection of the molecules in collision) obtained from these trajectories are plotted as a function of b/bmax in Fig. 10. The solid lines drawn correspond to those for the Lennard-Jones potential, VLJ(r) = 4εLJ [(σLJ/r)12 − (σLJ/r)6]; the angle of classical elastic scattering for an isotropic potential, V(r), is given as follows (see, e.g., ref 72):

= E − 2 r

d

;1 − /IJK − /

(34)

where r0 is the distance of closest approach, which is given as the largest real root of the equation 1 − V(r0)/Erel − b2/r02 = 0. The angles calculated for the Lennard-Jones potential closely correspond to the trajectory results, which indicates that the Lennard-Jones parameters used are consistent with the potential energy surface employed in the trajectory calculations, and that the intermolecular dynamics of the collisions can be approximately described by isotropic interactions. To explore potential correlations between the intermolecular dynamics and the collision frequency, the effective impact parameter, beff, is artificially defined here as beff/bmax = (Z1/Ztraj)1/2 or, in an explicit form (see eqs 24 and 25),

33



ln

〈∆d ln ∆d 〉RI S J = }2 + ln〈∆d 〉RI S − 〈∆d 〉RI S max

Page 34 of 57

(35)

which gives beff/bmax =0.55 and 0.46 for Erel = 5 and 20 kJ mol−1, respectively. These values of b/bmax are within the range in which collisions tend to result in negative scattering angles. A straightforward interpretation of this observation is that the energy transfer effectively occurs via repulsive interactions, whereas most trajectories that trace only the attractive part of the interaction force lead to quasi-elastic collisions. This interpretation motivates the use of a capture rate constant as the collision frequency for the exponential-down model. The dashed lines in Fig. 10 represents angles calculated using eq 34 with a hypothetical potential that describes only the attractive term, Vdisp(r) = −4εLJ (σLJ/r)6. The angle decreases sharply at b = bcap, where bcap is the impact parameter below which trajectories are captured by the attractive force, and is given as72 1/'

cap = 31/ ^pq IJK /lpq

(36)

For the cases of Erel = 5 and 20 kJ mol−1, bcap/bmax = 0.57 and 0.45, respectively, both of which coincide closely with the values of beff/bmax. The capture rate constant is determined from the corresponding cross-section and the distribution of Erel given by eq 7 as follows:

cap = 2 Γ2/3MN O/lpq

1/

E^LJ ;8MN O/E[

(37)

An identical result can be obtained using canonical variational transition-state theory. Figure 11 shows a comparison of Z1 with ZLJ and Zcap as a function of kBT/εLJ. To facilitate the comparison, they are plotted relative to the hard-sphere collision frequency for the diameter

σLJ, which is defined by LJHS = E^LJ ;8MN O/E[

(38)

34


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The ratios of Z1 to ZLJHS for all the systems investigated in the present study were found to be effectively characterized as a function of kBT/εLJ, and are well represented by Zcap/ZLJHS. Considering that ZLJ has been commonly employed in master equation models for unimolecular reactions merely as a matter of practical convenience, Zcap should certainly serve as a better reference quantity than ZLJ if the exponential-down model is employed for P(E, Eʹ). The present discussion is based on the trajectory results for monatomic bath gases. The method proposed in the present study for determining an appropriate collision frequency from trajectory-based energy transfer moments should also be applicable to polyatomic colliders. Additional work is needed to investigate the collision frequency for energy transfer in polyatomic bath gases. In particular, it would be worthwhile to examine whether capture rate constants can generally be reasonable representations of the collision frequency for various types of intermolecular forces, including dispersion and dipole-dipole interactions.

35



Figure 10. Angles of the scattering in collisions of C2H6 with Ar at Evib = 400 kJ mol−1, Trot = 1000 K, and Erel = 5 and 20 kJ mol−1 (dots). The elastic scattering angles for the isotropic potentials, VLJ and Vdisp, are shown as solid and dashed lines, respectively.

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Figure 11. Comparison between the collision frequencies Z1 (symbols), ZLJ (dashed line), and Zcap (solid line) as a function of kBT/εLJ.

CONCLUSIONS The present study proposes a robust and consistent method for deriving the collision frequency for energy transfer using classical trajectory calculations. The method has been applied to the collisions of several types of polyatomic molecules with monatomic bath gases. The results and discussion lead to the following conclusions: (i) Collision frequencies that are appropriate for modeling collisional energy transfer depend on an assumed functional form of the energy transfer probability. (ii) The conventional Lennard-Jones model underestimates the collision frequency, at least for the energy transfer probability functions considered. This conclusion is in accordance with the implications of the previous studies. (iii) The limiting low-pressure rate constant is sensitive to energy transfer moments near 37



the first-order but neither to those of the high-orders nor to those that arise from quasi-elastic collisions. This justifies the use of the exponential-down model, together with an appropriate collision frequency, in practical master equation applications for single-channel unimolecular reactions. (iv) Collision frequencies that are suitable for the exponential-down model can be approximately represented by the capture rate constants.

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Collision Frequency for Energy Transfer in Unimolecular Reactions

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