Comparison of the rate constants of a bimolecular reaction using two methods

3.1 Stationary points

The optimized geometric parameters of the reactants, products, intermediates and transition states involved in CH₃Cl + OH reaction at the MP2/6-311+G (d,p) level of theory are presented in Fig. 1 along with experimental geometries (Malcolm and Chase, 1998; Malcolm et al., 1985; William et al., 1956; Escribano et al., 1996) and theoretical ones of Theodora et al. (2005). The harmonic vibrational frequencies of the intermediates and transition states are shown in Table 1 (see Table 2 in Ref. Maryam and fahimeh (2012) for reactants and products). Two channels involved in the CH₃Cl with OH reaction can be described as the following:

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Figure 1

Optimized geometries of all the reactants, intermediates, transition states and products for the reaction CH₃Cl + OH at the MP2/6-311+G (d,p). Bond lengths are in angstroms and angles are in degrees. ^aTheodora et al. (2005), ^bMalcolm and Chase (1998), ^cMalcolm et al. (1985), ^dSander et al. (2002), ^eWilliam et al. (1956) and ^fEscribano et al. (1996). (See above mentioned references for further information.)

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The major problem in the application of the unrestricted calculation is that of contamination with higher spin states. The expectation values of S² range from 0.779 to 0.755 and from 1.00 to 0.759 before annihilation at the UMP2/6-311+G (d,p) level. After annihilation, S² is from 0.750 to 0.750 and from 0.760 to 0.750 for the reactions (12) and (13), respectively. Therefore the spin contamination is not severe.

When the OH radical is getting closer to the hydrogen atom of the CH₃Cl molecule, the transition state TS1 is located, which connects the complex IM1 to the complex IM2 in order to form products (CH₂Cl + H₂O) in the reaction channel (12). In the reaction channel (13) a Cl atom migrates via TS2 to yield IM4 and products (CH₃ + HOCl). In the H-abstraction reaction, as shown in Fig. 1, the O–H4 bond is shortened by 1.32 Å and it arrives to the O-H4 bond in H₂O (product) and the C–H4 bond is lengthened by 0.67 Å and in the Cl-abstraction reaction the O–Cl bond is shortened by 0.62 Å and it arrives to the O–Cl bond in HOCl (product) and the C–Cl bond is lengthened by 1.04 Å.

Table 2 shows the relative energies including zero-point energies (ZPEs) of the stationary points at different levels of theory with respect to the reactants energy. In order to obtain more reliable energies for rate constant calculation, we have used the G2MP2 method (see Table 1 in Ref. Maryam and Fahimeh (2012)).

3.2 Properties of the reaction path

The classical potential energy (V_MEP), the ground-state vibrational adiabatic potential energy $(V_{\text{a}}^{\text{G}})$ and the zero-point energy (ZPE) for reactions channels (13) as functions of the intrinsic reaction coordinates at the MP2/6-311+G (d, p) are plotted Fig. 2, where $V_{\text{a}}^{\text{G}}(s)=V_{\text{MEP}}(s)+\text{ZPE}(s)$ (for reaction (12) see Fig. 4 in Ref Maryam and Fahimeh (2012)). Fig. 2 shows that the position of the maximum of $V_{\text{MEP}}(s)$ is almost the same as that of $V_{\text{a}}^{\text{G}}(s)$ , and the ZPE curve is constant in the vicinity of the saddle point. To analyze this behavior in more detail, we have presented the variations of generalized mode vibrational frequencies along the MEP for this reaction. Table 3 indicate the vibrational frequencies and the vibrational modes assignment. In Fig. 3, in the negative limit of s (s = −∞), the frequencies correspond to the reactants, while in the positive limit of s (s = +∞), the frequencies are associated with the products (see Fig. 2 in Ref Fahimeh and Maryam (2012) for reaction (12)). In Fig. 3, modes 7, 8 and 9 indicate the CH₃ twist; rock in plane and rock out of plane in IM3 and it drops linearly after the saddle point and goes to zero for products, this decrease shows Cl has been added to the oxygen atom. Mode 10 represents the C–Cl stretch which their frequency increases in IM1 region and after the saddle point becomes linear. The mode shown by the solid line 11, which relates to the breaking C–Cl bond in the reactant region (the C–Cl stretch mode) and the forming Cl–O bond in the product region (the C–Cl–O stretch mode), changes sharply in the region from s = −1.5–1.0 (amu)^1/2 bohr which relates to the Cl–O stretch mode. In addition, in the saddle point (s = 0), TS2, four whole modes (rock in plane, bend in plane, rock out of plane and twist) and C–Cl–O stretch mode at 871i denote the formation of the Cl–O bond. Thus the imaginary frequency is relative to the forming bond Cl–O and broking bond C–Cl. The two lowest frequencies (14 and 15 modes, especially 15) have a maximum near the saddle point and tend to be zero for reactants and products. These changes should cause large decreases in the ZPE (Fig. 2). These comparisons show that the reaction (13) has completely performed. The other frequencies do not change significantly in going from the reactants to products.

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Figure 2

Classical potential energy (V_MEP), vibrational adiabatic potential energy $(V_{\text{a}}^{\text{G}})$ and ZPE as functions of the intrinsic reaction coordinate, s, at the G2MP2 method for the reaction (13).

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Figure 3

Change of the generalized mode vibrational frequencies for the reaction (13) as functions of the reaction coordinate, s, at the MP2/6-311+G (d,p).

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Comparison of modes in Table 3 shows that transition state has a reactant-like character in the reaction channel (12), it means this reaction is exothermic (Fahimeh and Maryam, 2012) and for the reaction channel (13) transition state has a product-like character, it means this reaction is endothermic. Furthermore, In order to decide whether transition states have a product- or reactant-like character, the following parameter (L) was used (Marie-Therese et al., 1993):

The changes of the bond length along the MEP for the reaction CH₃Cl + OH as functions of s are described in Fig. 4 for the reaction channel (13). It appears that for this reaction the active O–Cl and C–Cl bonds change strongly in the course of the reaction and the other bond lengths are almost invariant during the reaction process(see Fig. 3 in Ref Fahimeh and Maryam (2012) for reaction (15)).

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Figure 4

The changing of some key bond distances along the reaction coordinate for the reaction (13) at the MP2/6-311+G (d,p) level.

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3.3 Rate constants

For the two-channel reaction CH₃Cl + OH, the individual channel rate constants, k₁ and k_{2 are} calculated using the transition state theory (Eq. (7)) incorporating the Winger tunneling correction and the hindered rotor approximation at the G2MP2 method in the temperature range of 200–2000 K. The results are given in Table 4 with experimental results (Sander et al., 2002).The total rate constant k_t for this reaction is calculated as the sum of the corresponding individual rate constants, that is, k_t = k₁ + k₂. From Tables 4 and 5 we can see that in the temperature range of 200–2000 K, k₁ is larger than k₂, and the total rate constant k_t is equal to the rate constant of the H-abstraction reaction k₁, thus the H-abstraction reaction is a major reaction channel in the temperature range of 200–2000 K. Fig. 5 shows the branching fractions k₁/k_t and k₂/k_t are dependent on the temperature.

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Figure 5

The logarithm of calculated branching ratio for the reaction CH₃Cl + OH as a function of 10³/T.

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As shown in Fig. 6(a) and (b), the tunneling correction plays an important role in the lower temperature range in the H-abstraction reaction. The TST-HR/W rate constants for the reaction channel (12) are significantly larger than the TST rate constants in the temperature range of 200–1000 K and they are asymptotic to the TST ones at higher temperatures. For example, ratios k (TST /W) to k (TST) are 9.11, 4.65, 3.02, 1.50, 1.14 and 1.08 at 200, 298, 400, 800, 1500 and 2000 K, respectively. In addition, the k (TST-HR/TST) ratios are 2.82, 2.47, 2.21, 1.65, 1.23 and 1.07 at 200, 298, 400, 800, 1500 and 2000 K, respectively. Thus, the tunneling correction and the hindered rotor approximation to the calculation of rate of hydrogen abstraction are significant in the low temperatures, but the tunneling correction is important in the low temperatures. For the Cl-abstraction reaction, (13), the ratios k (TST/W) to k (TST) and the k (TST-HR/TST) ratios fall within 2.63–1.01 and 2.07–0.71 over the temperature range of 200–2000 K. In comparison with H-abstraction reaction, the tunneling correction is small for the Cl-abstraction reaction. Furthermore, Fig. 6 shows that rate constants are nearly the same in the whole temperature range. Finally, the calculated rate constants for the major channel can be expressed, in cm³ molecule⁻¹ s⁻¹, by the following fits: $$k=6.91\times {10}^{-11}{T}^{-0.61}{e}^{\frac{-1290.05}{T}}\text{for temperature range}200–400\text{K and}$$ $$k=9.11\times {10}^{-11}{T}^{-0.01}{e}^{\frac{-3576.25}{T}}\text{for temperature range}400–2000\text{K.}$$

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Figure 6

(a) The logarithm of calculated rate constants as a function of 10³/T for the reaction (12). ^aTheodora et al. (2005) and ^bSander et al. (2002). (b) Same as those in (a) except for the reaction (13).

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Because the dominant channel is formation of CH₂Cl and H₂O, therefore rate constants are calculated with another method. For this channel, the reaction mechanism can be processed with the following pathway:

According to formalism of Eq (9), the rate constant for the multi-step H-abstraction from CH₃Cl by OH radical, can be calculated where Q_A and Q_B are the partition functions of CH₃Cl and OH, respectively. The bottom limit of the integral is given by the threshold energy of the reaction. The sums of states are calculated using the Beyer–Swinehart algorithm (Stephan et al., 1973; Dieter et al., 1979). Results of the rate constant calculations for the reaction under investigation are given in Table 6. From Table 6, one can see that rate constants using the equations developed by Jerzy et al. are closer to the experimental results (Sander et al., 2002).The profiles obtained at the G2MP2 level are displayed in Fig. 7 for this reaction. The reaction pathway is abstraction of a Hydrogen atom in the CH₃ group of IM1 to produce IM2 via TS4. Attack of OH radical on CH₃Cl molecule leads, in the first elementary step, to formation of the hydrogen-bonded molecular complex IM1, and next via the transition state TS1 to the molecular complex IM2, which then dissociates into products (CH₂Cl + H₂O). This profile shows both IM1 and IM2 complexes are located below the reactant energy level and IM1 is more unstable.

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Figure 7

Schematic energy profile of the potential energy surface for the reaction (12) at the G2MP2 level.

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