Anharmonic effect of the unimolecular dissociation of CH3COOH

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Anharmonic effect of the unimolecular dissociation of

CH

₃

COOH

Liwei Zhanga, Li Yaoa, Qian Lia, Guiqiu Wanga & S.H. Linb a

Department of Physics, Dalian Maritime University, Dalian, China b

Department of Applied Chemistry, National Chiao-Tung University, Hsin-chu, Taiwan Published online: 12 May 2014.

To cite this article: Liwei Zhang, Li Yao, Qian Li, Guiqiu Wang & S.H. Lin (2014) Anharmonic effect of the unimolecular

dissociation of CH3COOH, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 112:21, 2853-2871, DOI: 10.1080/00268976.2014.915066

To link to this article: http://dx.doi.org/10.1080/00268976.2014.915066

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RESEARCH ARTICLE

Anharmonic effect of the unimolecular dissociation of CH

COOH

a_{Department of Physics, Dalian Maritime University, Dalian, China;}b_{Department of Applied Chemistry, National Chiao-Tung} University, Hsin-chu, Taiwan

(Received 17 June 2013; accepted 10 April 2014)

Anharmonic and harmonic rate constants of the reactions have been calculated with the Rice–Ramsperger–Kassel–Marcus theory, and the anharmonic results are higher than the harmonic ones. The anharmonic effect and isotopic effect on the decomposition reactions have also been examined. The anharmonic effect in all the four reactions is obvious, especially at the high temperatures or energies. Relatively, the anharmonic effect on the reaction trans-acetic acid→ TS2 radical is the least obvious among the four reactions. In the microcanonical system, the difference of the rate constants between the deuterated results and the non-deuterated results is not negligible; in other words, the isotopic effect is obvious in all the four reactions. Among these reactions, three of the deuterated results (d, d3 and d4) have a big difference.

Keywords: anharmonic effect; unimolecular reaction; rate constant; RRKM method; isotopic effect

1. Introduction

Acetic acid is a very important material of organic synthe-sis, which is often used for the synthesis to produce vinyl acetate and acetic anhydride. There are two reaction mech-anisms for the gas-phase unimolecular decomposition of acetic acid: the dehydration process leading to water and ketene, and the decarboxylation, leading to methane and carbon dioxide,

CH3COOH→ H2O+ CH2CO, (1) CH3COOH→ CH4+ CO2. (2) Bamford and Dewar measured the activation energies in a flow system for both the two channels at 1068–1218 K and got activation energies of 67.5 kcal mol−1 for dehy-dration and 62.0 kcal mol−1for decarboxylation [1]. Blake and Jackson studied these reactions in both batch and flow system and concluded that the decomposition was of the first order when temperature was above 1000 K [2,3], and the activation energy was 64.9 kcal mol−1for this process. All these authors considered that a possible reason from a bimolecular mechanism might cause an artificially low activation energy in the unimolecular process and thus sug-gested that the 67.5 kcal mol−1activation energy measured by Bamford and Dewar should be more realistic [4]. Blake and Jackson got an activation energy of 58.5 kcal mol−1 in a batch system for decarboxylation reaction and an ac-tivation energy of 69.8 kcal mol−1 measured in a flow

∗

Corresponding author. Email: [email protected]

system [2,3]. Mackie and Doolan surveyed this decom-position reaction at 1300–1950 K in a shock tube and they got the same result. This decomposition occurred via two competing unimolecular channels (reactions 1 and 2), and both of the activation energies were 72.7 kcal mol−1 [5]. Thus, activation energies for reaction 1 were in the range 67.5–72.7 kcal mol−1, while for reaction 2, the activation energies were in the range 64.9–72.7 kcal mol−1[4].

However, the activation energies which were given by theoretical studies were higher than experimental ones [6–9]. Decarboxylation (reaction 2) was predicted to have a higher activation energy than that for dehydration [4]. The reaction barrier of the decarboxylation process was 89.3 kcal mol−1, which was proceeded with the MP4/6-31G∗ and HF/6-31G∗ methods [6]. With the MP4/6-31G∗ and HF/6-31G∗ methods, the calculations for the dehydration process obtained an activation barrier of 81.3 kcal mol−1 and an activation barrier of 80.6 kcal mol−1for a two-step dehydration process: isomerisation to an enediol via hydro-gen transfer as Equation (3), followed by water elimination via a four-centre transition state as Equation (4) [4,9],

CH3COOH→ CH2C(OH)2, (3) CH2C(OH)→ H2O+ CH2CO. (4) This paper mainly studies activation energies, and rate constants for the title reactions, the anharmonic effect and isotopic effect have also been discussed. Many chemists

C

2014 Taylor & Francis

Table 1. The energetic parameters of the reactant and transition states, obtained from MP2/6-311++ G (3df,2p) calculations.

Cis-acetic acid Enediol Trans-acetic acid TS1 TS2 TS3 TS4

Zero-point energy (MP2) H 0.06232 0.06124 0.06209 0.05431 0.05409 0.05660 0.05583 (Hartree) d 0.05886 0.05791 0.05865 0.05111 0.05223 0.05313 0.05369 d3 0.05267 0.05169 0.05244 0.04610 0.04475 0.04843 0.04630 d4 0.04920 0.04835 0.04900 0.04289 0.04288 0.04496 0.04415 Imaginary frequencies H – – – –1811 –1980 –2113 –1629 d –1808 –1525 –2113 –1221 d3 –1337 –1960 –1553 –1628 d4 –1334 –1503 –1553 –1220 Single-point energy (CCSD(t)) H –228.7505 –228.7057 –228.7423 –228.6228 –228.6284 –228.6290 –228.6288 (Hartree) d d3 d4

Barrier (kcal mol−1) H – – – 75.19 66.42 72.64 44.73

d 75.24 67.39 72.60 45.62

d3 76.05 66.58 73.61 44.87

d4 76.18 67.65 73.61 45.59

had observed that the anharmonic effect was consider-able in molecular systems, especially in those of molecules and clusters with highly flexible transition states [10]. In 1962, Schlag and Sandsmark discovered that anharmonic-ity corrections might be significant in practice [11]. Then, Haarhoff considered the anharmonic corrections about the density of vibrational energy levels were for a system of simple Morse oscillators [12]. At present, many chemists have also observed that anharmonic effects are quite event-ful in many dissociation of clusters and molecular systems [13–17]. It is recognised that the characteristic features of the anharmonic effect contain an increase in the bond lengths and distance of bond dissociation, and a decrease in vibrational bond-stretching frequencies [13–17]. Several authors have concentrated on the requirement for the anhar-monic correction to previous reaction rate theories [18–20]. For this reason, using experimental thermodynamic data, Troe presented a simple empirical method for generating anharmonic vibrational densities of states [21]. Recently, a method proposed by Yao and Lin (YL) [22] could carry out the first principle calculations about the rate constants of molecular reactions within the framework of the transition state theory proposed. With this method, the anharmonic ef-fect that was on the dissociation of molecular reaction had been examined. The results suggested that the YL method was appropriate to calculate rate constants of the unimolec-ular reaction and investigating the anharmonic effect of rate constant.

To our knowledge, although many efforts have been made to research the unimolecular dissociation of CX3COOX (X = H, D), the study of the rate constant and the anharmonic effect on the reaction is still very rare. The purpose of this paper is to calculate the harmonic and

anharmonic rate constants of the unimolecular dissociation of CX3COOX (X= H, D) and investigate the anharmonic effect on the reactions using YL method. In addition, in this paper, isotopic effect is also given which is divided into three aspects, deuterated out on the methyl hydrogens (d3), on the carboxyl hydrogens (d) and on all the hydro-gens in acetic acid molecule (d4). The energetic parameters of the reactant and the transition are in accordance with the theoretical results given in Ref. [4]. The computational methods will be described in detail in Section 2. Section 3 concludes numerical results and detailed discussions. Fi-nally, concluding remarks and a summary of this work are presented in Section 4.

2. Computational methods

2.1. Ab initio calculations

This article contains four unimolecular dissociations included in Equations (5)–(7), as following:

cis-acetic acid → TS1 → CH2CO+ H2O, (5)

trans-acetic acid → TS2 → CH4+ CO2, (6)

cis-acetic acid → TS3 → enediol → TS4

→ CH2CO+ H2O. (7)

The geometry optimisations of the reactant CH3COOH and transition states have been accomplished by the MP2 method with 6−311 + + G(3df,2p) basis set. The charac-terisation of stationary points, zero-point energy (ZPE) cor-rections, as well as the calculations of reaction rate constant

within the framework of the transition state theory (TST) and the Rice–Ramsperger–Kassel–Marcus (RRKM) theory are performed by the vibrational harmonic and anharmonic frequencies. All stationary points have been positively iden-tified as local minima or transition states. The Gaussian 03 program is utilised for all ab initio calculations [23].

For the microcanonical system with energy E, in the RRKM theory, the unimolecular reaction rate constant can be expressed as [24]

k(E) =σ h

W=(E − E=)

ρ(E) , (8)

where σ is the symmetry factor (here, we set σ = 1), h is Planck’s constant, ρ (E) stands for the density of the state of the reactant, W=(E) represents the total number of states for the transition state, and E and E=are the total energy and the activation energy in the microcanonical case, respectively. From the definition of W (E) and ρ (E), we can obtain [25–27] W (E) = i H (E − Ei), (9) ρ(E) = dW (E) dE , (10)

where H (E − Ei) denotes Heaviside function, Ei, are

en-ergy levels.

The Laplace transformation is employed for W (E) and

ρ (E) , we obtain [25–27]  _∞ 0 dEe −βE_{W (E) =} Q(β) β = L[W(E)], (11)  _∞ 0

dEe−βEρ(E) = Q(β) = L[ρ(E)], (12)

where β = 1kT , k is Boltzmann’s constant, T is the

tem-perature of the system and Q (β) is the partition function of the system. That is, approximate W (E) and ρ (E) can be determined by Equations (11) and (12) by applying the inverse Laplace transformation after Q (β) are given.

For a canonical system, the rate constant k (T ) for uni-molecular reaction can be expressed as [25,27–29]

k(T ) = kT h Q=(T ) Q(T ) e −E= kT , (13)

where Q (T ) and Q=(T ) are the partition functions of the reactant and the activated complex, respectively. In this case, we get Q=(T ) = N−1_ i qi=(T ), (14) _Tab le 2 . T he rate constants from cis -acetic acid (H) to TS1 (H) at dif ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 Ener gy (kcal mol − 1) 13.62 27.11 42.29 58.37 68.30 74.99 81.74 88.53 95.36 102.22 109.11 Cor respond (cm − 1) 4763.60 9481.72 14,790.93 20,414.91 23,887.93 26,227.75 28,588.57 30,963.37 33,352.16 35,751.45 38,161.22 Anhar monic rate constant (s − 1) 1.09 × 10 − 3 8.55 × 10 2 9.31 × 10 5 6.75 × 10 7 4.28 × 10 8 1.19 × 10 9 2.90 × 10 9 6.33 × 10 9 1.26 × 10 10 2.30 × 10 10 3.93 × 10 10 Har monic rate constant (s − 1) 4.84 × 10 − 4 2.24 × 10 2 1.60 × 10 5 8.37 × 10 6 4.59 × 10 7 1.18 × 10 8 2.71 × 10 8 5.63 × 10 8 1.08 × 10 9 1.93 × 10 9 3.27 × 10 9

Figure 1. The canonical and microcanonical rate constants for cis-acetic (H) acid to TS1 (H). The unit of rate constant is s−1. Q(T ) = N  i qi(T ), (15)

where N is the number of the vibrational modes of the reac-tant, qi=(T ) and qi(T ) are the vibrational partition function

of the transition state and the reactant for each mode. The discussions above show that the partition func-tion is considerable in the calculafunc-tion of k (E) and k (T ). Recently, the quantum chemical calculations can provide the information of cubic and quartic anharmonic poten-tial functions not only for polyatomic molecules but also for clusters. We have applied these anharmonic potential functions to treat the intramolecular vibrational redistribu-tions (or relaxaredistribu-tions) of polyatomic molecules and clusters and solved the corresponding Schr¨odinger equations per-turbatively to the second-order approximations which give us the same accuracy as given by Equation (16). This ap-proach has been employed in this paper for the calculation of W=(E − E0) and ρ (E) in Ref. [30]. In this work, to calculate the partition function, Morse Oscillator (MO) is often used: Eni =  ni+ 1 2  ωi− xi  ni+ 1 2 2 ωi, (16)

where xi is the MO parameter, and it can be expressed as

xi = _4Dω_iei, in which Die represents the well depth of MO.

In our study, xi is obtained from anharmonic frequency

calculations in Gaussian 03. ωi is the frequency of the ith

vibrational mode, and ni is the vibration quantum

num-ber of the vibrational mode. The maximum value of ni is

represented by nmax

i . In this paper, we will discuss the

an-harmonic rate constant of the concerted exchange reaction in the CH3COOH and its isomer.

In the calculation of the density of states, E = −∂ ln Q

∂β

, harmonic and anharmonic degrees of freedom of the reactant are 18 (3N – 6, N= 8). To calculate the total number of states W (E), harmonic and anharmonic imagi-nary frequencies are excluded and the degrees of freedom of the transition state are 17.

3. Results and discussions

The energetic parameters of the reactant and transition states about the unimolecular dissociation of the acetic acid are collected in Table 1. The data are obtained with the MP2/6-311+ + G (3df,2p) method. To obtain high accu-racy and reliability, we recompute the single-point ener-gies at CCSD(T)/6-311G(d,p) level, and then get the values of the energy barrier 75.19, 72.64, 44.73 and 66.42 kcal mol−1, which are in accordance with the values obtained by Michael Page [4]. The results calculated with CCSD(T)/6-311+ + G(d,p) are used in the rate constants calculation.

3.1. Unimolecular dissociation of the cis-acetic

acid radical (the channel which includes the TS1)

For this dissociation, the anharmonic and harmonic rate constants (H) for canonical case are presented inTable 2, with temperatures ranging from 1000 to 4000 K. From Table 2, it is clear that both the harmonic and anharmonic rate constants increase with the temperatures increasing.

Table 3. The rate constants from cis-acetic acid (H) to TS1 (H) at different energies for the microcanonical system. The unit of rate constant is s−1.

Temperature (K) 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

Energy (kcal mol−1) 78.36 81.74 85.13 88.53 91.94 95.36 98.79 102.22 105.66 109.11

Correspond (cm−1) 27,406.41 28,588.57 29,774.22 30,963.37 32,156.02 33,352.16 34,551.80 35,751.45 36,954.59 38,161.22 Anharmonic rate constant (s−1) 2.67× 103 _{3.29× 10}4 _{2.11× 10}5 _{9.34× 10}5 _{3.23× 10}6 _{9.38× 10}6 _{2.39× 10}7 _{5.47× 10}7 _{1.15× 10}8 _{2.27× 10}8 Harmonic rate constant (s−1) 1.49× 103 _{1.56× 10}4 _{8.80× 10}4 _{3.50× 10}5 _{1.10× 10}6 _{2.95× 10}6 _{6.97× 10}6 _{1.49× 10}7 _{2.93× 10}7 _{5.42× 10}7

Table 4. The rate constants from cis-acetic acid-D to TS1-D at different temperatures for the canonical system. The unit of rate constant is s−1. Temperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 d Anharmonic rate Constant (s−1) 1.18× 10−3 8.99× 103 _{9.22× 10}5 _{6.27× 10}7 _{3.85× 10}8 _{1.05× 10}9 _{2.54× 10}9 _{5.49× 10}9 _{1.09× 10}10 _{1.98× 10}10 _{3.41× 10}10 Harmonic rate constant (s−1) 4.96× 10−4 2.28× 102 _{1.62× 10}5 _{8.47× 10}6 _{4.64× 10}7 _{1.19× 10}8 _{2.73× 10}8 _{5.68× 10}8 _{1.09× 10}9 _{1.95× 10}9 _{3.29× 10}9 d3 Anharmonic rate constant (s−1) 5.84× 10−4 4.91× 102 _{5.30× 10}5 _{3.80× 10}7 _{2.42× 10}8 _{6.79× 10}8 _{1.68× 10}9 _{3.73× 10}9 _{7.58× 10}9 _{1.43× 10}10 _{2.51× 10}10 Harmonic rate constant (s−1) 2.87× 10−4 1.51× 102 _{1.13× 10}5 _{6.10× 10}6 _{3.38× 10}7 _{8.74× 10}7 _{2.01× 10}8 _{4.19× 10}8 _{8.06× 10}8 _{1.45× 10}9 _{2.45× 10}9 d4 Anharmonic rate constant (s−1) 6.21× 10−4 5.55× 102 _{6.19× 10}5 _{4.41× 10}7 _{2.76× 10}8 _{7.64× 10}8 _{1.85× 10}9 _{4.04× 10}9 _{8.04× 10}9 _{1.48× 10}10 _{2.54× 10}10 Harmonic rate constant (s−1) 2.82× 10−4 1.50× 102 _{1.13× 10}5 _{6.07× 10}6 _{3.36× 10}7 _{8.70× 10}7 _{2.00× 10}8 _{4.17× 10}8 _{8.02× 10}8 _{1.44× 10}9 _{2.44× 10}9

Table 5. The rate constants from cis-acetic acid-D to TS1-D at different energies for the microcanonical system. The unit of rate constant is s−1. d Energy (kcal mol−1) 76.36 79.75 83.15 86.56 89.98 93.41 96.84 100.29 103.74 107.19 110.65 Correspond (cm−1) 26,706.91 27,892.56 29,081.71 30,274.36 31,470.51 32,670.15 33,869.79 35,076.43 36,283.07 37,489.70 38,699.84 Anharmonic rate constant (s−1) 1.88× 102 _{5.86× 10}3 _{5.59× 10}4 _{3.15× 10}5 _{1.29× 10}6 _{4.27× 10}6 _{1.20× 10}7 _{2.98× 10}7 _{6.72× 10}7 _{1.39× 10}8 _{2.71× 10}8 Harmonic rate constant (s−1) 1.10× 102 _{2.77× 10}3 _{2.31× 10}4 _{1.16× 10}5 _{4.31× 10}5 _{1.30× 10}6 _{3.36× 10}6 _{7.74× 10}6 _{1.62× 10}7 _{3.16× 10}7 _{5.76× 10}7 d3 Energy (kcal mol−1) 79.12 82.54 85.97 89.42 92.86 96.32 99.78 103.25 106.72 110.2 113.68 Correspond (cm−1) 27,672.22 28,868.37 30,068.01 31,274.65 32,477.79 33,687.92 34,898.06 36,111.69 37,325.32 38,542.45 39,759.58 Anharmonic rate constant (s−1) 4.06× 102 _{6.70× 10}3 _{5.21× 10}4 _{2.67× 10}5 _{1.03× 10}6 _{3.28× 10}6 _{8.98× 10}6 _{2.18× 10}7 _{4.85× 10}7 _{9.97× 10}7 _{1.92× 10}8 Harmonic rate constant (s−1) 2.21× 102 _{3.06× 10}3 _{2.08× 10}4 _{9.55× 10}4 _{3.36× 10}5 _{9.87× 10}5 _{2.51× 10}6 _{5.74× 10}6 _{1.20× 10}7 _{2.32× 10}7 _{4.23× 10}7 d4 Energy (kcal mol−1) 77.04 80.48 83.93 87.38 90.84 94.31 97.79 101.27 104.75 108.24 111.73 Correspond (cm−1) 26,944.74 28,147.88 29,354.52 30,561.16 31,771.29 32,984.92 34,202.05 35,419.18 36,636.31 37,856.94 39,077.57 Anharmonic rate constant (s−1) 1.39× 101 _{7.37× 10}2 _{9.37× 10}3 _{6.41× 10}4 _{3.04× 10}5 _{1.13× 10}6 _{3.50× 10}6 _{9.44× 10}6 _{2.28× 10}7 _{5.05× 10}7 _{1.04× 10}8 Harmonic rate constant (s−1) 9.91 4.08× 102 _{4.47× 10}3 _{2.70× 10}4 _{1.15× 10}5 _{3.88× 10}5 _{1.10× 10}6 _{2.74× 10}6 _{6.14× 10}6 _{1.27× 10}7 _{2.43× 10}7

Figure 2. The canonical and microcanonical rate constants (anharmonic) for cis-acetic-D acid to TS1-D. The unit of rate constant is s−1.

Corresponding toTable 2, the rate constants (H) for the reaction are plotted in Figure 1. With the temperatures increasing from 1000 to 4000 K, the harmonic rate constants change from 4.84× 10−4 to 3.27× 109 s−1, while the anharmonic rate constants are in the range from 1.09× 10−3 to 3.93× 1010 s−1. The gap between anhar-monic rate constants and haranhar-monic ones changes with the increasing temperatures. When the temperature is 1000 K, the anharmonic rate constant (1.09× 10−3 s−1) is 55.6%

higher than the harmonic rate constant (4.84× 10−4 s−1), and the anharmonic rate constant (3.93× 1010s−1) is 91.7% higher than the harmonic rate constant (3.27× 109 s−1) at 4000 K. It is worth noting the first six energy data in Table 2, as they are all lower than the calculated activation energy 75.19 kcal mol−1. Hence, we have to calculate the rate constants in the microcanonical system at higher energy.

Figure 3. The canonical and microcanonical rate constants (harmonic) for cis-acetic-D acid to TS1-D. The unit of rate constant is s−1.

Table 6. The rate constants from cis-acetic acid (H) to TS3 (H) at different temperatures for the canonical system. The unit of rate constant is s−1. Temperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 Energy (kcal mol−1) 13.62 27.11 42.29 58.37 68.30 74.99 81.74 88.53 95.36 102.22 109.11 Correspond (cm−1) 4763.60 9481.72 14,790.93 20,414.91 23,887.93 26,227.75 28,588.57 30,963.37 33,352.16 35,751.45 38,161.22 Anharmonic rate constant (s−1) 8.96× 10−4 3.15× 102 _{2.15× 10}5 _{1.17× 10}7 _{6.70× 10}7 _{1.78× 10}8 _{4.20× 10}8 _{9.04× 10}8 _{1.79× 10}9 _{3.33× 10}9 _{5.82× 10}9 Harmonic rate constant (s−1) 5.99× 10−4 1.50× 102 _{7.79× 10}4 _{3.37× 10}6 _{1.70× 10}7 _{4.18× 10}7 _{9.21× 10}7 _{1.84× 10}8 _{3.43× 10}8 _{5.97× 10}8 _{9.84× 10}8

Figure 4. The canonical and microcanonical rate constants for cis-acetic (H) acid to TS3 (H). The unit of rate constant is s−1.

To calculate the energy corresponding to the above tem-peratures, we employ the relation between the total energy of a microcanonical system and the temperatures of a canon-ical system by E = − ∂ ln Q ∂β  (17)

with Equation (17), and the energy in the microcanoni-cal system can be obtained. Then the total energies are

27,406.41 to 38,161.22 cm−1, corresponding to the tem-peratures of 3100 to 4000 K, respectively, for this channel. Table 3shows the harmonic and anharmonic rate constants (H) of the title reaction obtained with the YL method for microcanonical case.

FromTable 3andFigure 1, we can see that for the mi-crocanonical system, the harmonic and anharmonic rate constants increase with the increasing of the total en-ergies. With the energies increasing from 27,406.41 to 38,161.22 cm−1, the harmonic rate constants increase from

Table 7. The rate constant from cis-acetic acid (H) to TS3(H) at different energies for the microcanonical system. The unit of rate constant is s−1. Energy (kcal mol−1) 74.99 78.36 81.74 85.13 88.53 91.94 95.36 98.79 102.22 105.66 109.11 Correspond (cm−1) 26,227.75 27,406.41 28,588.57 29,774.22 30,963.37 32,156.02 33,352.16 34,551.80 35,751.45 36,954.59 38,161.22 Anharmonic rate constant (s−1) 9.60× 102 _{1.05× 10}4 _{6.00× 10}4 _{2.40× 10}5 _{7.60× 10}5 _{2.05× 10}6 _{4.86× 10}6 _{1.05× 10}7 _{2.09× 10}7 _{3.90× 10}7 _{6.89× 10}7 Harmonic rate constant (s−1) 6.24× 102 _{6.22× 10}3 _{3.35× 10}4 _{1.28× 10}5 _{3.91× 10}5 _{1.02× 10}6 _{2.34× 10}6 _{4.88× 10}6 _{9.42× 10}6 _{1.71× 10}7 _{2.93× 10}7

T ab le 8 . T he rate constants from cis -acetic acid-D to T S3-D at dif ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 d A nhar monic rate Constant (s − 1) 1.02 × 10 − 3 3.59 × 10 2 2.47 × 10 5 1.37 × 10 7 7.97 × 10 7 2.15 × 10 8 5.16 × 10 8 1.13 × 10 9 2.28 × 10 9 4.30 × 10 9 7.66 × 10 9 Har monic rate constant (s − 1) 6.10 × 10 − 4 1.51 × 10 2 7.86 × 10 4 3.40 × 10 6 1.71 × 10 7 4.21 × 10 7 9.26 × 10 7 1.86 × 10 8 3.45 × 10 8 6.00 × 10 8 9.88 × 10 8 d3 Anhar monic rate constant (s − 1) 4.54 × 10 − 4 1.87 × 10 2 1.41 × 10 5 8.39 × 10 6 5.02 × 10 7 1.37 × 10 8 3.33 × 10 8 7.34 × 10 8 1.49 × 10 9 2.82 × 10 9 5.02 × 10 9 Har monic rate constant (s − 1) 3.33 × 10 − 4 9.6 × 10 1 5.30 × 10 4 2.36 × 10 6 1.20 × 10 7 2.98 × 10 7 6.59 × 10 7 1.33 × 10 8 2.47 × 10 8 4.31 × 10 8 7.13 × 10 8 d4 Anhar monic rate constant (s − 1) 4.72 × 10 − 4 1.90 × 10 2 1.38 × 10 5 7.85 × 10 6 4.61 × 10 7 1.24 × 10 8 3.01 × 10 8 6.59 × 10 8 1.33 × 10 9 2.52 × 10 9 4.48 × 10 9 Har monic rate constant (s − 1) 3.33 × 10 − 4 9.60 × 10 1 5.29 × 10 4 2.36 × 10 6 1.20 × 10 7 2.98 × 10 7 6.58 × 10 7 1.32 × 10 8 2.47 × 10 8 4.31 × 10 8 7.12 × 10 8 T ab le 9 . T he rate constants from cis -acetic acid-D to T S3-D at dif ferent ener g ies for the m icrocanonical system. T he unit o f rate constant is s − 1. d E ner g y (kcal mol − 1) 72.98 76.36 79.75 83.15 86.56 89.98 93.41 96.84 100.29 103.74 107.19 110.65 Cor respond (cm − 1) 25,524.76 26,706.91 27,892.56 29,081.71 30,274.36 31,470.51 32,670.15 33,869.79 35,076.43 36,283.07 37,489.70 38,699.84 Anhar monic rate constant (s − 1) 7.96 × 10 1 2.09 × 10 3 1.78 × 10 4 9.01 × 10 4 3.37 × 10 5 1.03 × 10 6 2.70 × 10 6 6.29 × 10 7 1.34 × 10 7 2.66 × 10 7 4.93 × 10 7 8.70 × 10 7 Har monic rate constant (s − 1) 5.96 × 10 1 1.15 × 10 3 9.04 × 10 3 4.34 × 10 4 1.55 × 10 5 4.52 × 10 5 1.14 × 10 6 2.56 × 10 6 5.23 × 10 6 1.00 × 10 7 1.80 × 10 7 3.06 × 10 7 d3 Ener gy (kcal mol − 1) 75.70 79.12 82.54 85.97 89.42 92.86 96.32 99.78 103.25 106.72 110.20 Cor respond (cm − 1) 26,476.08 27,672.22 28,868.37 30,068.01 31,274.65 32,477.79 33,687.92 34,898.06 36,111.69 37,325.32 38,542.45 Anhar monic rate constant (s − 1) 1.21 × 10 2 1.92 × 10 3 1.36 × 10 4 6.42 × 10 4 2.33 × 10 5 6.96 × 10 5 1.81 × 10 6 4.23 × 10 6 9.03 × 10 6 1.79 × 10 7 3.34 × 10 7 Har monic rate constant (s − 1) 7.80 × 10 1 1.11 × 10 3 7.37 × 10 3 3.28 × 10 4 1.13 × 10 5 3.23 × 10 5 8.07 × 10 5 1.80 × 10 6 3.70 × 10 6 7.04 × 10 6 1.26 × 10 7 d4 Ener gy (kcal mol − 1) 77.04 80.48 83.93 87.38 90.84 94.31 97.79 101.27 104.75 108.24 111.73 115.23 Cor respond (cm − 1) 26,944.74 28,147.88 29,354.52 30,561.16 31,771.29 32,984.92 34,202.05 35,419.18 36,636.31 37,856.94 39,077.57 40,301.69 Anhar monic rate constant (s − 1) 2.36 × 10 2 2.72 × 10 3 1.68 × 10 4 7.25 × 10 4 2.47 × 10 5 7.12 × 10 5 1.80 × 10 6 4.09 × 10 6 8.54 × 10 6 1.67 × 10 7 3.07 × 10 7 5.37 × 10 7 Har monic rate constant (s − 1) 1.55 × 10 2 1.65 × 10 3 9.69 × 10 3 4.00 × 10 4 1.31 × 10 5 3.62 × 10 5 8.83 × 10 5 1.94 × 10 6 3.91 × 10 6 7.39 × 10 7 1.31 × 10 7 2.23 × 10 7

T ab le 10. The rate constants from enediol (H) to T S4 (H) at d if ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 Ener gy (kcal mol − 1) 46.56 49.74 52.95 56.19 59.45 62.73 66.03 69.34 72.68 76.02 Cor respond (cm − 1) 16,284.36 17,396.57 18,519.26 19,652.45 20,792.64 21,939.82 23,093.99 24,251.67 25,419.83 26,588.00 Anhar monic rate constant (s − 1) 6.17 × 10 8 1.08 × 10 9 1.81 × 10 9 2.94 × 10 9 4.59 × 10 9 6.98 × 10 9 1.03 × 10 10 1.49 × 10 10 2.11 × 10 10 2.93 × 10 10 Har monic rate constant (s − 1) 1.82 × 10 8 3.03 × 10 8 4.81 × 10 8 7.35 × 10 8 1.09 × 10 9 1.56 × 10 9 2.18 × 10 9 2.97 × 10 9 3.98 × 10 9 5.21 × 10 9 T ab le 11. The rate constants from enediol (H) to T S4 (H) at d if ferent ener gies for the microcanonical system. T he unit o f rate constant is s − 1. Ener gy (kcal mol − 1) 46.56 49.74 52.95 56.19 59.45 62.73 66.03 69.34 72.68 76.02 Cor respond (cm − 1) 16,284.36 17,396.57 18,519.26 19,652.45 20,792.64 21,939.82 23,093.99 24,251.67 25,419.83 26,588.00 Anhar monic rate constant (s − 1) 7.13 × 10 4 7.91 × 10 5 4.22 × 10 6 1.55 × 10 7 4.46 × 10 7 1.09 × 10 8 2.38 × 10 8 4.70 × 10 8 8.67 × 10 8 1.50 × 10 9 Har monic rate constant (s − 1) 5.20 × 10 4 5.11 × 10 5 2.54 × 10 6 8.80 × 10 6 2.42 × 10 7 5.64 × 10 7 1.17 × 10 8 2.20 × 10 8 3.87 × 10 8 6.39 × 10 8

1.49× 103 _{to 5.42× 10}7 _s−1_{, while the anharmonic rate} constants are in the range from 2.67× 103_{to 2.27× 10}8_s−1_. The values of harmonic rate constants and the anhar-monic ones increase with the increasing energies. When the total energy is 27,406.41 cm−1, the anharmonic rate constant (2.67× 103 s−1) is 44.2% higher than the har-monic rate constant (1.49× 103s−1), while the anharmonic rate constant (2.27× 108 s−1) is 76.1% higher than the harmonic rate constant (5.42× 107 s−1) at total energy 38,161.22 cm−1.

InTable 4and5, isotopic effect results of this reaction are also given. The rate constants of the isotopic effect cases are illustrated inFigures 2and3. In the canonical case, as a whole, the isotopic effect is not obvious, and the result of the deuterated anharmonic rate constants by size is H > d > d4 > d3 (at the temperatures higher than 2000 K), for example, when temperature is 2000 K, the anhar-monic rate constant for CH3COOH case is 9.31× 105s−1, the anharmonic rate constant for CH3COOD case is 9.22× 105s−1, for CD3COOD case is 6.19× 105 s−1and for CD3COOH case is 5.30× 105s−1. The deuterated har-monic rate constants by size is d > H > d3 > d4, when tem-perature is 1000 K, the harmonic rate constant for d case is 4.96× 10−4s−1, the harmonic rate constant for CH3COOH case is 4.84× 10−4 s−1, for CD3COOH case is 2.87× 10−4s−1and for CD3COOD case is 2.82× 10−4s−1. In the microcanonical system, the isotopic effect is obvious, both the result of the deuterated harmonic rate constants and the result of the deuterated harmonic rate constants by size are H > d > d3 > d4, which can be reached fromFigures 2 and3specifically.

3.2. Unimolecular dissociation of the cis-acetic

acid radical (the channel includes the TS3)

For this reaction, the anharmonic and harmonic rate con-stants (H) for canonical case are presented inTable 6, with temperatures ranging from 1000 to 4000 K. Corresponding to Table 6, the rate constants (H) for this reaction are plotted in Figure 4. From Table 6, it is clear that both the harmonic and anharmonic rate constants increase with temperatures increasing from 1000 to 4000 K. The harmonic rate constants increase from 5.99× 10−4 to 9.84× 108 _s−1_{, while the anharmonic rate constants} are in the range from 8.96× 10−4 to 5.82× 109 _s−1 _in Table 6. The anharmonic rate constants are higher than the harmonic ones at all the temperatures we calculated. When the temperature is 1000 K, the harmonic rate constant (5.99× 10−4s−1) is 33.1% lower than the anharmonic rate constant (8.96× 10−4 s−1), while the harmonic rate con-stant (9.84× 108_s−1_{) is 83.1% lower than the anharmonic} rate constant (5.82× 109 _s−1_{) at temperature 4000 K.} The first several energies are worth noting in Table 6as they are all lower than the calculated activation energy of

72.64 kcal mol−1 . Hence, we have to calculate the rate constants in the microcanonical system at higher energies.

The total energies are 26,227.75–38,161.22 cm−1, cor-responding to the temperatures of 3000–4000 K using Equation (17), respectively. Table 7 shows the harmonic and anharmonic rate constants (H) of the reaction obtained from the YL method for the microcanonical case. The rate constants of the microcanonical total energies are also il-lustrated inFigure 4.

FromTable 7andFigure 4, we can see that in the micro-canonical system the harmonic and anharmonic rate con-stants increase with the increasing of the total energies. With the energies increasing from 26,227.75 to 38,161.22 cm−1, the harmonic rate constants increase from 6.24× 102 _to 2.93× 107_s−1_{, while the anharmonic rate constants are in} the range from 9.60× 102 _{to 6.89× 10}7 _s−1_{. The} anhar-monic rate constants are higher than the haranhar-monic ones at all ranges of energy. When total energy is 26,227.75 cm−1, the harmonic rate constant (6.24× 102 _s−1_{) is 35.0%} lower than the anharmonic rate constant (9.60× 102 s−1), while the harmonic rate constant (2.93× 107s−1) is 57.5% lower than the anharmonic rate constant (6.89× 107s−1) at total energy 38,161.22 cm−1.

InTable 8and9, the isotopic effect results of this reac-tion are also given. The rate constants for the isotopic effect cases are illustrated inFigures 5and6. In the canonical case, as a whole, the isotopic effect is not obvious and the result of the deuterated anharmonic rate constants by size is d > H > d3 > d4 (the temperatures are higher than 2000 K). For example, when temperature is 2000 K, the anhar-monic rate constant for CH3COOD case is 2.47× 105s−1, the anharmonic rate constant for CH3COOH case is 2.15× 105 _s−1_{, for CD}

3COOH case is 1.41× 105s−1 and for CD3COOD case is 1.38× 105 s−1; the result of the deuterated harmonic rate constants by size is d > H > d3 > d4, when temperature is 2000 K, the harmonic rate con-stant for CH3COOD case is 7.86× 104s−1, the harmonic rate constant for CH3COOH case is 7.79× 10−4 s−1, for CD3COOH case is 5.30× 10−4 s−1 and for CD3COOD case is 5.29× 10−4s−1. In the microcanonical system, the isotopic effect is obvious and both the results of the ated harmonic rate constants and the results of the deuter-ated harmonic rate constants by size are H > d > d3 > d4, which can be reached fromFigures 5and6specifically.

3.3. Unimolecular dissociation of the enediol

radical (the channel includes the TS4)

For this reaction, the anharmonic and harmonic rate constants for canonical case are presented inTable 10, with temperatures ranging from 2100 to 3000 K. Corresponding to Table 10, the rate constants for the title reaction are plotted in Figure 7. From Table 10, it is clear that both the harmonic and anharmonic rate constants increase

Figure 5. The canonical and microcanonical rate constants (anharmonic) for cis-acetic-D acid to TS3-D. The unit of rate constant is s−1.

with temperatures increasing from 2100 to 3000 K. The harmonic rate constants increase from 1.82× 108 _to 5.21× 109_s−1_{, while the anharmonic rate constants are in} the range from 6.17× 108 _{to 2.93× 10}10 _s−1_{. The} anhar-monic rate constants are higher than the haranhar-monic ones at all range of temperatures. When the temperature is 2100 K, the harmonic rate constant (1.82× 108s−1) is 70.5% lower than the harmonic rate constant (6.17× 108s−1), while the harmonic rate constant (5.21× 109 s−1) is 82.2% lower than the anharmonic rate constant (2.93× 1010_s−1_{) at the} temperature 3000 K.

The total energies are 16,284.36–26,588.00 cm−1, cor-responding to the temperatures of 2100–3000 K using

Equation (17). Table 11shows the harmonic and anhar-monic rate constants of the title reaction obtained with the YL method for microcanonical case. The rate constants of the microcanonical total energies are also illustrated in Figure 7.

From Table 11 andFigure 7, we can see that for the microcanonical system the harmonic and anharmonic rate constants increase with the increase in total energy. With the energy increasing from 16,284.36 to 26,588.00 cm−1, the harmonic rate constants increase from 5.20× 104 _to 6.39× 108 _s−1_{, while the anharmonic rate constants are} in the range from 7.13× 104_{to 1.50× 10}9_s−1_{. The} anhar-monic rate constants are higher than the haranhar-monic ones at all

Figure 6. The canonical and microcanonical rate constants (harmonic) for cis-acetic-D acid to TS3-D. The unit of rate constant is s−1.

Figure 7. The canonical and microcanonical rate constants for enediol (H) to TS4 (H). The unit of rate constant is s−1.

range of energy. When the total energy is 16,284.36 cm−1, the harmonic rate constant (5.20× 104 _s−1_{) is 27.1%} lower than the anharmonic rate constant (7.13× 104 _s−1_), while the harmonic rate constant (6.39× 108_s−1_{) is 57.4%} lower than the anharmonic rate constant (1.50× 109_s−1_{) at} total energy 26,588.00 cm−1.

InTable 12and13, isotopic effect results of this effect are also given. The rate constants for the isotopic effect cases are illustrated inFigures 8and9. For the canonical case, the isotopic effect is obvious, and it is supposed that C– O bond scission has an effect on it. The result of the deuter-ated anharmonic rate constants by size is d3 > H > d4 > d. However, compared with the results of H, there is little

difference among the results of CD3COOD; however, the difference between the results of CH3COOH and the results of CH3COOD is in the range from 13.79% to 15.02%, and the results of H are about 43.16% to 44.63% lower than the deuterated results of CD3COOH case. For example, when the temperature is 2100 K, the anharmonic rate constant for CD3COOH case is 1.06× 109s−1, and the anharmonic rate constant for CH3COOH case is 6.17× 108 s−1, for CD3COOD case is 6.07× 108s−1and for CH3COOD case is 5.22× 108_s−1_{, The result of the deuterated harmonic rate} constants by size is H > d3 > d > d4; there is little difference among the results of the harmonic rate constants between the results of CH3COOH and CD3COOH case, nevertheless

Figure 8. The canonical and microcanonical rate constants (anharmonic) for enediol-D to TS4-D. The unit of rate constant is s−1.

Figure 9. The canonical and microcanonical rate constants (harmonic) for enediol-D to TS4-D. The unit of rate constant is s−1.

the difference between the results of CH3COOH case and CH3COOD case is in the range from 28.41% to 31.78%, The results of CH3COOH and the deuterated results of CD3COOD case is in the range from 28.41% to 31.35%. For example, when the temperature is 2100 K, the harmonic rate constant for CH3COOH is 1.82× 108s−1, and the har-monic rate constant for CD3COOH case is 1.76× 108s−1, for CH3COOD case is 1.24× 10−4s−1and for CD3COOD case is 1.24× 10−4s−1. In the microcanonical system, both the result of the deuterated harmonic rate constants and the result of the deuterated harmonic rate constants by size are H > d3 > d > d4, this conclusion can be reached from Figures 8and9specifically.

3.4. Unimolecular dissociation of the trans-acetic

acid radical (the channel includes the TS2)

This reaction, the anharmonic and harmonic rate constants for canonical case are presented in Table 14, with tem-peratures ranging from 1000 to 4000 K. Corresponding to Table 14, the rate constants for the title reaction are plotted inFigure 10. FromTable 14, it is clear that both the harmonic and anharmonic rate constants increase with the increasing temperatures. With the temperatures increasing from 1000 to 4000 K, the harmonic rate constants increase from 0.43× 10−2to 8.91× 109_s−1_{, while the anharmonic} rate constants are in the range from 0.52× 10−2 to 1.66× 1010s−1. The anharmonic rate constants are higher

Figure 10. The canonical and microcanonical rate constants for trans-acetic acid (H) to TS2 (H). The unit of rate constant is s−1.

Figure 11. The canonical and microcanonical rate constants (anharmonic) for trans-acetic acid-D to TS2-D. The unit of rate constant is s−1.

than the harmonic ones at all range of temperatures. When the temperature is 1000 K, the harmonic rate constant (0.43× 10−2 s−1) is 17.3% lower than the harmonic rate constant (0.52× 10−2 s−1), while the harmonic rate con-stant (8.91× 109_s−1_{) is 46.3% lower than the anharmonic} rate constant (1.66× 1010 _s−1_{) at the temperature 4000 K.} The first several energies are worth noting inTable 14as they are all lower than the calculated activation energy of 66.42 kcal mol−1. Hence, we have to calculate the rate constants in the microcanonical system at higher energies.

The total energies are 23,940.39–37,007.05 cm−1, cor-responding to the temperatures from 2800 to 3900 K, respectively, using Equation (17).Table 15shows the har-monic and anharhar-monic rate constants of the title reaction obtained with the YL method for microcanonical case. The rate constants of the microcanonical total energies are also illustrated inFigure 10.

FromTable 15andFigure 10, for the microcanonical system, we can see that the harmonic and anharmonic rate constants increase with the increasing total energies. With

Figure 12. The canonical and microcanonical rate constants (harmonic) for trans-acetic acid-D to TS2-D. The unit of rate constant is s−1.

T ab le 12. The rate constants from enediol-D to TS4-D at d if ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 d A nhar monic rate constant (s − 1) 5.22 × 10 8 9.21 × 10 8 1.55 × 10 9 2.52 × 10 9 3.95 × 10 9 6.01 × 10 9 8.88 × 10 9 1.28 × 10 10 1.80 × 10 10 2.49 × 10 10 Har monic rate constant (s − 1) 1.24 × 10 8 2.08 × 10 8 3.33 × 10 8 5.12 × 10 8 7.61 × 10 8 1.10 × 10 9 1.54 × 10 9 2.11 × 10 9 2.84 × 10 9 3.73 × 10 10 d3 Anhar monic rate constant (s − 1) 1.06 × 10 9 1.90 × 10 9 3.23 × 10 9 5.27 × 10 9 8.29 × 10 9 1.26 × 10 10 1.86 × 10 10 2.68 × 10 10 3.76 × 10 10 5.16 × 10 10 Har monic rate constant (s − 1) 1.76 × 10 8 2.92 × 10 8 4.65 × 10 8 7.11 × 10 8 1.05 × 10 9 1.51 × 10 9 2.12 × 10 9 2.89 × 10 9 3.87 × 10 9 5.07 × 10 10 d4 Anhar monic rate constant (s − 1) 6.07 × 10 8 1.07 × 10 9 1.82 × 10 9 2.95 × 10 9 4.63 × 10 9 7.04 × 10 9 1.04 × 10 10 1.50 × 10 10 2.11 × 10 10 2.92 × 10 10 Har monic rate constant (s − 1) 1.24 × 10 8 2.08 × 10 8 3.32 × 10 8 5.11 × 10 8 7.61 × 10 8 1.10 × 10 9 1.54 × 10 9 2.11 × 10 9 2.83 × 10 9 3.73 × 10 10 T ab le 13. The rate constants from enediol-D to TS4-D at d if ferent ener gies for the microcanonical system. T he unit o f rate constant is s − 1. d E ner g y (kcal mol − 1) 47.55 50.77 54.02 57.29 60.58 63.89 67.22 70.56 73.92 77.29 – – Cor respond (cm − 1) 16,630.61 17,756.81 18,893.50 20,037.18 21,187.86 22,345.53 23,510.20 24,678.36 25,853.52 27,032.18 Anhar monic rate constant (s − 1) 5.22 × 10 4 6.09 × 10 5 3.40 × 10 6 1.29 × 10 7 3.85 × 10 7 9.66 × 10 7 2.14 × 10 8 4.30 × 10 8 8.00 × 10 8 1.40 × 10 9 –– Har monic rate constant (s − 1) 2.67 × 10 4 2.77 × 10 5 1.45 × 10 6 5.19 × 10 6 1.47 × 10 7 3.52 × 10 7 7.44 × 10 7 1.43 × 10 8 2.55 × 10 8 4.26 × 10 8 –– d3 Ener gy (kcal mol − 1) 46.48 49.74 53.03 56.33 59.66 63.00 66.36 69.74 73.12 76.52 79.93 – Cor respond (cm − 1) 16,256.38 17,396.57 18,547.24 19,701.42 20,866.09 22,034.25 23,209.41 24,391.57 25,573.72 26,762.87 27,955.52 Anhar monic rate constant (s − 1) 2.14 × 10 4 3.72 × 10 5 2.52 × 10 6 1.08 × 10 7 3.53 × 10 7 9.55 × 10 7 2.26 × 10 8 4.81 × 10 8 9.40 × 10 8 1.72 × 10 9 2.98 × 10 9 – Har monic rate constant (s − 1) 9.88 × 10 3 1.48 × 10 5 9.19 × 10 5 3.65 × 10 6 1.12 × 10 7 2.83 × 10 7 6.28 × 10 7 1.26 × 10 8 2.31 × 10 8 4.00 × 10 8 6.50 × 10 8 – d4 Ener gy (kcal mol − 1) 47.44 50.74 54.06 57.40 60.76 64.14 67.53 70.93 74.34 77.77 81.20 115.23 Cor respond (cm − 1) 16,592.14 17,746.32 18,907.49 20,075.65 21,250.81 22,432.97 23,618.62 24,807.77 26,000.42 27,200.06 28,399.70 40,301.69 Anhar monic rate constant (s − 1) 1.28 × 10 4 2.13 × 10 5 1.44 × 10 6 6.21 × 10 6 2.04 × 10 7 5.56 × 10 7 1.32 × 10 8 2.79 × 10 8 5.45 × 10 8 9.93 × 10 8 1.71 × 10 9 5.37 × 10 7 Har monic rate constant (s − 1) 6.11 × 10 3 9.03 × 10 5 5.68 × 10 5 2.32 × 10 6 7.21 × 10 6 1.87 × 10 7 4.21 × 10 7 8.54 × 10 7 1.59 × 10 8 2.77 × 10 8 4.56 × 10 8 2.23 × 10 7

T ab le 14. The rate constants from tr ans -acetic acid (H) to TS2(H) at dif ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 Ener gy (kcal mol − 1) 13.74 27.25 42.43 58.52 68.45 75.14 81.89 88.68 95.51 102.37 109.26 Cor respond (cm − 1) 4805.57 9530.69 14,839.89 20,467.37 23,940.39 26,280.22 28,641.03 31,015.83 33,404.62 35,803.91 38,213.69 Anhar monic rate constant (s − 1) 0.52 × 10 − 2 6.25 × 10 3 2.31 × 10 6 8.14 × 10 7 3.75 × 10 8 8.74 × 10 8 1.83 × 10 9 3.52 × 10 9 6.26 × 10 9 1.05 × 10 10 1.66 × 10 10 Har monic rate constant (s − 1) 0.43 × 10 − 2 4.20 × 10 3 1.38 × 10 6 4.54 × 10 7 2.04 × 10 8 4.72 × 10 8 9.83 × 10 8 1.88 × 10 9 3.34 × 10 9 5.60 × 10 9 8.91 × 10 9 T ab le 15. The rate constants from tr ans -acetic acid (H) to TS2 (H) at dif ferent ener g ies for the m icrocanonical system. T he unit o f rate constant is s − 1. Ener gy (kcal mol − 1) 68.45 71.79 75.14 78.51 81.89 85.28 88.68 92.09 95.51 98.93 102.37 105.81 Cor respond (cm − 1) 23,940.39 25,108.55 26,280.22 27,458.87 28,641.03 29,826.68 31,015.83 32,208.48 33,404.62 34,600.77 35,803.91 37,007.05 Anhar monic rate constant (s − 1) 2.35 × 10 3 3.55 × 10 4 2.35 × 10 5 1.02 × 10 6 3.39 × 10 6 9.38 × 10 6 2.26 × 10 7 4.91 × 10 7 9.78 × 10 7 1.81 × 10 8 3.17 × 10 8 5.28 × 10 8 Har monic rate constant (s − 1) 2.66 × 10 3 3.44 × 10 4 2.05 × 10 5 8.27 × 10 5 2.61 × 10 6 6.90 × 10 6 1.61 × 10 7 3.38 × 10 7 6.57 × 10 7 1.19 × 10 8 2.05 × 10 8 3.36 × 10 8

T ab le 16. The rate constants from tr ans -acetic acid-D to T S2-D at dif ferent temperatures for the canonical system. T he unit o f rate constant is s − 1. T emperature (K) 1000 1500 2000 2500 2800 3000 3200 3400 3600 3800 4000 d A nhar monic rate constant(d1) (s − 1) 0.33 × 10 − 1 5.12 × 10 3 2.23 × 10 6 8.90 × 10 7 4.36 × 10 8 1.06 × 10 9 2.29 × 10 9 4.53 × 10 9 8.29 × 10 9 1.42 × 10 10 2.31 × 10 10 Har monic rate constant(d1) (s − 1) 0.25 × 10 − 1 2.81 × 10 3 9.87 × 10 5 3.37 × 10 7 1.54 × 10 8 3.58 × 10 8 7.49 × 10 8 1.44 × 10 9 2.57 × 10 9 4.32 × 10 9 6.89 × 10 9 d3 Anhar monic rate constant(d3) (s − 1) 0.49 × 10 − 1 5.75 × 10 3 2.07 × 10 6 7.11 × 10 7 3.22 × 10 8 7.41 × 10 8 1.54 × 10 9 2.91 × 10 9 5.12 × 10 9 8.46 × 10 9 1.33 × 10 10 Har monic rate constant(d3) (s − 1) 0.42 × 10 − 1 4.12 × 10 3 1.36 × 10 6 4.50 × 10 7 2.02 × 10 8 4.68 × 10 8 9.74 × 10 8 1.86 × 10 9 3.31 × 10 9 5.55 × 10 9 8.84 × 10 9 d4 Anhar monic rate constant(d4) (s − 1) 0.30 × 10 − 1 4.48 × 10 3 1.78 × 10 6 6.36 × 10 7 2.91 × 10 8 6.75 × 10 8 1.41 × 10 9 2.67 × 10 9 4.72 × 10 9 7.83 × 10 9 1.23 × 10 10 Har monic rate constant(d4) (s − 1) 0.22 × 10 − 1 2.65 × 10 3 9.44 × 10 5 3.25 × 10 7 1.49 × 10 8 3.47 × 10 8 7.27 × 10 8 1.39 × 10 9 2.50 × 10 9 4.20 × 10 9 6.72 × 10 9 T ab le 17. The rate constants from tr ans -acetic acid-D to T S2-D at dif ferent ener g ies for the m icrocanonical system. T he unit o f rate constant is s − 1. d E ner g y (kcal mol − 1) 69.73 73.09 76.47 79.86 83.26 86.68 90.10 93.53 96.96 100.41 103.86 107.31 Cor respond (cm − 1) 24,388.07 25,563.23 26,745.38 27,931.04 29,120.19 30,316.33 31,512.48 32,712.12 33,911.76 35,118.40 36,325.04 37,531.67 Anhar monic rate constant (s − 1) 1.71 × 10 3 2.58 × 10 4 1.77 × 10 5 8.01 × 10 5 2.77 × 10 6 7.97 × 10 6 1.99 × 10 7 4.45 × 10 7 9.13 × 10 7 1.75 × 10 8 3.15 × 10 8 5.39 × 10 8 Har monic rate constant (s − 1) 1.72 × 10 3 2.20 × 10 4 1.34 × 10 5 5.54 × 10 5 1.78 × 10 6 4.80 × 10 6 1.13 × 10 7 2.41 × 10 7 4.72 × 10 7 8.66 × 10 7 1.50 × 10 8 3.91 × 10 8 d3 Ener gy (kcal mol − 1) 69.06 72.45 75.86 79.27 82.7 86.13 89.57 93.02 96.48 99.94 103.40 106.88 Cor respond (cm − 1) 24,153.74 25,339.39 26,532.04 27,724.68 28,924.33 30,123.97 31,327.11 32,533.75 33,743.88 34,954.02 36,164.15 37,381.28 Anhar monic rate constant (s − 1) 6.48 × 10 2 1.11 × 10 4 8.18 × 10 4 3.84 × 10 5 1.37 × 10 6 4.02 × 10 6 1.02 × 10 7 2.32 × 10 7 4.80 × 10 7 9.22 × 10 7 1.66 × 10 8 2.84 × 10 8 Har monic rate constant (s − 1) 8.64 × 10 2 1.25 × 10 4 8.29 × 10 4 3.64 × 10 5 1.24 × 10 6 3.49 × 10 6 8.60 × 10 6 1.90 × 10 7 3.86 × 10 7 7.30 × 10 7 1.30 × 10 8 2.19 × 10 8 d4 Ener gy (kcal mol − 1) 70.32 73.74 77.16 80.60 84.05 87.51 90.97 94.44 97.91 101.39 104.88 108.37 Cor respond (cm − 1) 24,594.42 25,790.57 26,986.71 28,189.85 29,396.49 30,606.62 31,816.76 33,030.39 34,244.02 35,461.15 36,681.78 37,902.41 Anhar monic rate constant (s − 1) 3.96 × 10 2 7.12 × 10 3 5.53 × 10 4 2.78 × 10 5 1.06 × 10 6 3.28 × 10 6 8.72 × 10 6 2.06 × 10 7 4.41 × 10 7 8.72 × 10 7 1.61 × 10 8 2.81 × 10 8 Har monic rate constant (s − 1) 4.87 × 10 2 7.29 × 10 3 4.99 × 10 4 2.27 × 10 5 7.92 × 10 5 2.29 × 10 6 5.75 × 10 6 1.29 × 10 7 2.65 × 10 7 5.07 × 10 7 9.12 × 10 7 1.56 × 10 8

the energies increasing from 23,940.39 to 37,007.05 cm−1, the harmonic rate constants increase from 2.66× 103 _to 3.36× 108_s−1_{, while the anharmonic rate constants are in} the range from 2.35× 103 _{to 5.28× 10}8 _s−1_{. The} anhar-monic rate constants are higher than the haranhar-monic ones except when the energy is 23,940.39 cm−1. When to-tal energy is 25,108.55 cm−1, the harmonic rate constant (3.44× 104 s−1) is 3.1% lower than the anharmonic rate constant (3.55× 104s−1), while the harmonic rate constant (3.36× 108_s−1_{) is 36.4% lower than the anharmonic rate} constant (5.28× 108_s−1_{) at total energy 37,007.05 cm}−1_{. It} can be seen that, the anharmonic effect is not obvious at the first several energies, but in the case of high energies that it is not negligible. In other words, the anharmonic effect is not such obvious as the other Three.

In Table 16 and17, isotopic effect results of this re-action are also given. The rate constants for the isotopic effect cases are illustrated inFigures 11 and12. For the canonical case, the isotopic effect is not obvious and the d4’s result of the deuterate anharmonic rate constants is the smallest among the four values, but the other three data have no regular size. The result of the deuterated har-monic rate constants by size is H > d3 > d > d4, for example, when temperature is 4000 K, the harmonic rate constant for CH3COOH case is 8.91× 109 s−1, the har-monic rate constant for CD3COOH case is 8.84× 109s−1, for CH3COOD case is 6.89× 109s−1 and for CD3COOD case is 6.72× 10−4s−1. In the microcanonical system, both the result of the deuterated harmonic rate constants and the result of the deuterated harmonic rate constants by size are H > d > d3 > d4, and this conclusion can be reached from Figures 11and12specifically.

For all the four unimolecular reactions, for the results of CH3COOH or the deuterated results, we can see that the harmonic and anharmonic rate constants increase with the temperatures and total energies increasing in the canonical or the microcanonical system. In other words, the anhar-monic effect becomes more significant with the tempera-tures and energies increasing, which cannot be neglected, while the anharmonic effect on the reaction trans-acetic acid→ TS2 radical is the smallest one. The rate constants of the deuterated results are smaller than the results of H for all the four reactions. In the results of the CH3COOH and the deuterated ones, the anharmonic effect becomes manifest along with the increase of temperatures or total energies. In the microcanonical system, the results of H are always the biggest ones among the deuterated rate constants of the four reactions.

Various systems and reactions give different results. We can find lots of examples in our previous papers [22]. The increasing ratio of total number of states W (E) and density of state ρ (E) are different. For anharmonic rate constants and harmonic rate constants, we cannot expect which one is higher than the other one.

4. Conclusions

In this paper, the anharmonic and harmonic rate constants have been calculated for canonical and microcanonical case of the four reactions with the YL method. The anharmonic effect on the decomposition reactions has been examined at MP2/6-311+ + G(3df,2p) level. All the values of barrier height (H) coincide with the earlier theoretical work [4] very well.

For the reaction cis-acetic acid→ TS1, we obtain the barrier height (H) 75.19 kcal mol−1, which is in accordance with the earlier theoretical (76.4 kcal mol−1)work by Page [4]. In both canonical and microcanonical cases, the rate constants increase with the total energies or the tempera-tures increasing basically. However, the difference between the two kinds of rate constants is notable. Therefore, we can draw the conclusion that the anharmonic effect is obvi-ous in either canonical or microcanonical system, and the deuterated results are also like this.

For the reaction cis-acetic acid→ TS3, we obtain the barrier height (H) 72.64 kcal/mol, which is in great agree-ment with the earlier theoretical (73.1 kcal mol−1) work by Page [4]. In both canonical and microcanonical cases, the rate constants increase with the total energies increasing basically. The harmonic and anharmonic rate constants are different at all the temperatures and energies. What is more, the difference increases with the total energies increasing. Therefore, we can draw conclusion that the anharmonic ef-fect is so significant that it cannot be neglected in either canonical or microcanonical system, also the CH3COOH results and the deuterated results.

For the reaction enediol→ TS4, we obtain the barrier height (H) 44.73 kcal mol−1, which is in reasonable agree-ment with the earlier theoretical (44.8 kcal mol−1) work in Ref. 4. In both canonical and microcanonical cases, the rate constants increase with the total energies increasing basi-cally, and the anharmonic rate constants are higher than those for harmonic cases, especially in the case of high total energies and temperatures, which indicates that the anharmonic effect of this reaction is significant. Therefore, we can draw a conclusion that the anharmonic effect is so significant that it cannot be neglected in either canonical or microcanonical system.

For the reaction trans-acetic acid→ TS2, we obtain the barrier height (H) 66.42 kcal mol−1, which is in reasonable agreement with the earlier theoretical (66.9 kcal mol−1) results in Ref. 4. In both canonical and microcanonical cases, the rate constants increase with the total energies in-creasing basically. However, the harmonic and anharmonic rate constants (H) produce similar results when the tem-peratures and energies are low, and the difference between the two kinds of rate constants is obvious at high energies and temperatures. What is more, the difference gradually becomes more and more obvious with the total energies increasing. Therefore, we can draw a conclusion that the

anharmonic effect is so significant that it is not negligi-ble in either canonical or microcanonical system, also the non-deuterated results.

For all the four reactions, compared with the non-deuterated results, the non-deuterated results differ from each other in the microcanonical system, and for the canonical system, except the reaction enediol → TS4, the isotopic effect of the four reactions is not obvious. As the C–H bond scission in each reaction is different, the three deuterated results on the different positions would result in the differ-ent deuterated results. Moreover, at differdiffer-ent temperatures, the deuterated results for each reaction are also different.

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11304028], [grant number 11304027]; the Natural Science Foundation of Liaoning Province [grant number 201102016]; the Fundamental Research Funds for the Central Universities [grant number 3132013102].

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