Stereodynamics Study of the Reaction of O(³P) with CH₄ (v = 0 ,j = 0)

Abstract

A new London-Eyring-Polanyi-Sato (LEPS) potential energy surface (PES) is used in the O + CH₄ → OH + CH₃ reaction via the quasiclassical trajectory method (QCT). Comparing with the experiments and the former ab initio calculations, the new LEPS PES describes the actual potential energy surface of the O + CH₄ reaction successfully. The four polarization dependent “generalized” differential cross sections (PDDCS) are presented in the center of mass frame. In the meantime, the distribution of dihedral angle [P(φ_r), the distribution of angle between k and j′ (P(θ_r)] and the angular distribution of product rotational vectors in the form of polar plots in θ_r and φ_r (P(θ_r, φ_r) are calculated. The isotope effect for the reactions O + CD₄ is also calculated. These results are in good agreement with the experiments.

1. Introduction

The reactions of ground-state atomic oxygen, O(³P), with hydrocarbons are the important initial steps of oxidation in combustion processes [1] and in Low Earth Orbit (LEO) conditions [2]. Among those reactions, the abstraction reaction O(³P) + CH₄ → OH + CH₃ has attracted considerable interest, both in experimental [3–7,11] and theoretical [8–10,12–20] calculations over the past decades. The rate constant values have been established in experiments [3,4]. Experimental studies have reported the translational energy release to the umbrella mode of the CH₃ product [5] as well as OH ro-vibrational distributions [6,7]. The thermal rate constants have been measured with a wide variety of methods [8–10]. The state distribution of the OH product was found by experiments [5,6,8,11] and theoretical [12] calculations. A small amount of OH rotational excitation was found [5,6,11], which has been interpreted as resulting from a direct abstraction mechanism with a preferentially collinear O-H-C approach of the O(³P) atom attacking to the C-H bond. Theoretical calculations [8,9,12,13] showed that this reaction has a collinear O-H-CH₃ transition state. The classical barrier height is around 0.4 eV [20,21]. Reduced-dimensionality quantum [16–19] models have been reported. The calculated kinetic behaviors and product state distributions are generally in good agreement with experimental findings [7].

It has been recognized that the correlated angular distribution provides an informative three-dimensional picture of a chemical reaction [22–25]. The angular distribution of the reagent and product relative velocity vector (k, k′) is characterized by the differential cross-section dσ/dω_t. Furthermore, the angular distribution describing the relative orientation of vectors k, k′ and product rotational quantum number j’ in space may be termed the k - k’ - j’ distribution. The correlations among three vectors in the center-of-mass frame can be characterized by certain interesting double and triple vector correlations [26]. To our best knowledge, there is only one experimental work [27] relate to the product angular distribution (k - k′), scalar and two-vector properties of the reaction were analyzed using the QCT method [14], the full-dimensional trajectory [13,20] calculations only relate to the k - k′ angular distributions. So the full product angular distribution of this reaction has not been reported.

We calculated the product rotational polarization, the scattering-angle resolved product rotational alignment, the vector correlations of the reaction O(³P) + CH₄ → OH + CH₃ and the isotope effect for the reactions O + CH₄/O + CD₄.

2. Results and Discussion

Figure 1 shows the minimum energy paths of the reaction O + CH₄ → OH + CH₃ at the collision energy of 0.65 eV from reactants to products on our chosen PESs. A new LEPS PES with a different set of Sato parameters has been calculated in our work. The Sato C parameters from [33] are on the PES1 and the Sato D parameters which we calculated are on the PES2. We get the Sato D parameters from the ab initio calculation carried out by Troya and García-Molina [20] when the quality of the minimum energy path on the new LEPS potential energy surface accords with the experimental results. In Figure 1, there is a potential barrier in the reaction. The values of the potential barrier are 11 kcal/mol on PES1 and 9.23 kcal/mol on PES2. The results on PES2 are accord with the calculation [20] in which the barrier is around 9.22 kal/mol. The product’s ro-vibrational distributions are shown in Figure 2. From Figure 2(a) we can see that the most probable vibrational quantum number of OH is v^' = 0. In Figure 2(b) the most probable rotational quantum number of OH is j′ = 1, which is quite close to the calculation [20] and experiment [37], but the product OH of our calculation is colder than the experimental results. The reason is that the collision energy changes to the product’s translational energy rather than rotational energy in the O+CH₄ direct abstraction reaction.

In the direct abstraction reaction, the rotational distribution is always in the narrow and low range of the rotational energy. The PDDCS (2π/σ)(dσ₀₀/dω_t), which is proportional to the differential cross section (DCS), predicts the angular distribution of the product molecular. In Figures 2(c,d) the DCS distribution is quite close to the experiment [27] and calculations [13,20]. The PES2 reflecting the real reaction process is much better than the PES1.

The value of (2π/σ)(dσ₂₀/dω_t), which is the expectation value of the second Legendre moment, shows the trend which is opposite to that of (2π/σ)(dσ₀₀/dω_t). At the extremes of forward and backward scattering, the PDDCSs with q ≠ 0 are necessarily zero. At these limiting scattering angles, the k -k’ scattering plane is not determined and the value of these PDDCSs with q ≠ 0 must be zero. The variations of the PDDCSs with k = 2 reflect changes in the rotational polarization with the scattering angles and suggest that the PDDCSs for the O+CH4 reaction contain important dynamical information. The four PDDCS of the OH product state are shown in Figure 3(a).

It indicates that the product OH scatters backward. The angular distribution that is asymmetric with θ_r = 90° is characteristic of a direct reaction mechanism. The available energy is released as product translation energy rather than internal excitation, and the product internal excitation is quite cold. This is consistent with Figure 2.

The P(θ_r) and P(φ_r) distributions are shown in Figures 3(b,c). We can get better graphical representation of the products polarization for the reaction O + CH₄. Figure 3(b) clearly shows that the distributions of the k - j’ correlation P(θ_r) peak at θ_r angle close to θ_r = 90° and is symmetric with θ_r = 90°. It demonstrates that j_OH is strongly aligned perpendicular to the line of centers. The distribution of the k - k’ - j’ correlation P(φ_r) is shown in Figure 3(c). The P(φ_r) tends to be asymmetric about φ_r = 180°, reflecting the strong polarization of angular momentum. There are two peaks of P(φ_r), respectivelyφ_r = 270°and φ_r = 90°. It implies that the angular momentum (j_OH) of the most product molecules aligns along the CM y-axis. This behavior suggests that the reaction proceeds preferentially when the reactant velocity vector lies in the plane containing all three atoms. However, for an initially random orientation of reactant molecules the probability of such planar collisions is very low; thus, we can conclude that the given PES reorients or polarizes the plane containing the three atoms into the k - k’ plane during the reaction process. The distribution of P(θ_r, φ_r) is presented in Figure 3(d); one peak appears at (90°, 270°). This suggests that the OH products are preferentially polarized perpendicular to the k - k’ plane. The P(θ_r, φ_r) distribution is not symmetric about φ_r=180°, reflecting the nonzero values of the PDDCS (2π/σ)(dσ₀₀/dω_t) for the O + CH4 reaction. It is in good accordance with the distribution of P(φ_r) that the dihedral angle distribution tends to be asymmetric with respect to the scattering plane.

Figure 4(a) presents the PDDCSs (2π/σ)(dσ₀₀/dω_t) distributions of the products OH and OD at the collision energy of 0.65eV. The distribution of the products OD is a little more backward than that of the products OH. The increase of backward scattering with the mass number indicates that the rotational angular momenta distribution of the products is sensitive to the factor of merit [the factor of merit is cos²β=m_Am_C/(m_A + m_B)(m_B + m_C) for the reaction A + BC → AB + C] [32,34].

The product polarization distributions for the reaction O + CH₄ and O + CD₄ are shown in Figures 4(b,c) that describe the visible stereodynamics isotope effect. Figure 4(b) illustrates that the product distribution of P(θ_r) for the O+ CD₄ reaction is a little broader than that for O + CH₄ reaction. This means that the rotational orientation effect of the product becomes weaker with the increase of the atomic mass. Han et al. [32,34]. have studied the product polarization for the reaction H + H’L (H, heavy; L, light), they found that the distribution of the product rotational angular momentum vectors is acutely sensitive to the mass factor, furthermore the increase of the mass factor can reduce the anisotropic distribution of the angular momentum j′ of the product molecule. The effect of mass factor cos²β = m_Am_C/(m_A + m_B)(m_B + m_C) on product rotational alignment is notable for the HHL mass combination reaction. The mass factor of the O + CD₄ reaction is larger than that of the O + CH₄ reaction. So the j’ tend toward a less anisotropic distribution with respect to the vector k, while the rotation of the product from the O + CH₄ is strongly aligned. During the reactive encounter, total angular momentum is conserved, j + L = j’ + L’ (here L and L’ are the reagent and product orbital angular momenta). According to the impulse model and j’ = Lsin²β + jcos²β + J₁m_B/m_AB, the larger product atom will take more angular momentum away, so the increase of the mass factor reduces the anisotropic distribution of j′.

Figure 4(c) describes that the P(φ_r) distribution becomes a little broader when the H atoms are displaced by the D atoms. With the increase of the atomic mass, the preference for in-plane reaction c gradually hanges to a preference for an out-of-plane mechanism. The mass effect causes the reaction O + CD4 to prefer an out-of-plane mechanism. That is to say, it is not necessary that the product molecules rotate in the scattering plane containing all the three atoms when the reaction occurs. According to the [35] and [36], the P(φ_r) is relevant to PES and the mass factor. We consider that the difference of the P(φ_r) distribution is probably attributed to the different mass number and to the isotope effect. The obvious variation in the dihedral distribution implies that the mass effect plays an important role in the dynamical stereochemistry.

3. Experimental Section

3.1. Product rotational polarization in the center-of-mass (CM) frame

The center-of-mass (CM) frame is chosen; in this frame, the z-axis is parallel to the reagent relative velocity k, and the y-axis is perpendicular to the xz-plane which contains k and k′. The distribution of the angular momentum j′ of the product molecule is described by the function f(θ_r), where θ_r is the angle between j’ and k. f(θ) can be represented by Legendre polynomial [26]:

$$f({\theta }_r)=\sum a_n{p}_n(\text{cos} {\theta }_r)$$

(1)

n = 2 indicates the product rotational alignment 〈p₂(j′ · k)〉 = 〈3 cos² θ_r − 1〉/2, where p₂ is the second Legendre moment, and the brackets indicate an average over the distribution of j’ about k.

The full three-dimensional angular distribution associated with k, k′ and j′ can be represented by a set of generalized polarization-dependent differential cross-sections (PDDCSs) in the CM frame. The k - k’ - j’ correlated CM angular distribution is written as the sum [26,29].

$$p({\omega }_t,{\omega }_r)=\sum _{kq}\frac{[k]}{4\pi }\frac{1}{\sigma }\frac{d{\sigma }_{kq}}{d{\omega }_t}{C}_{kq}({\theta }_r,{\phi }_r)*$$

(2)

where (1/σ)(dσ_kq/dω_t) is a generalized polarization-dependent differential cross-section (PDDCS), and (1/σ)(dσ_kq/dω_t) yields.

$$\begin{array}{cc}\frac{1}{\sigma }\frac{d{\sigma }_{k0}}{d{\omega }_t}=0& k \text{is} \text{odd},\end{array}$$

(3)

$$\frac{1}{\sigma }\frac{d{\sigma }_{kq+}}{d{\omega }_t}=\frac{1}{\sigma }\frac{d{\sigma }_{kq}}{d{\omega }_t}+\frac{1}{\sigma }\frac{d{\sigma }_{k-q}}{d{\omega }_t}=0, k\text{ is even}, q\text{ is odd or }k\text{ is odd}, q\text{ is even}.,$$

(4)

$$\frac{1}{\sigma }\frac{d{\sigma }_{kq-}}{d{\omega }_t}=\frac{1}{\sigma }\frac{d{\sigma }_{kq}}{d{\omega }_t}-\frac{1}{\sigma }\frac{d{\sigma }_{k-q}}{d{\omega }_t}=0, k\text{ is even}, q\text{ is odd or }k\text{ is odd}, q\text{ is odd}.$$

(5)

The PDDCS is written in the following form:

$$\frac{1}{\sigma }\frac{d{\sigma }_{kq\pm }}{d{\omega }_t}=\frac{1}{4\pi }\sum _{k_1}[k_1]S_{kq\pm }^{k_1}{C}_{k_1-q}({\theta }_t,0)$$

(6)

where the $S_{kq\pm }^{k_1}$ is evaluated using the expected value expression

$$S_{kq\pm }^{k_1}=\langle C_{k_{1}q}({\theta }_t,0)C_{kq}({\theta }_r,0)[{(-1)}^q{e}^{iq{\varphi }_r}\pm e^{-iq{\varphi }_r}]\rangle$$

(7)

where the angular brackets represent an average over all angles.

The PDDCS with q=0 is presented by:

$$\frac{1}{\sigma }\frac{d{\sigma }_{k0}}{d{\omega }_t}=\frac{1}{4\pi }\sum _{k_1}[k_1]S_{k0}^{k_1}{p}_{k_1}(\text{cos} {\theta }_t)$$

(9)

where $S_{k0}^{k_1}$ is evaluated by the expected value expression:

$$S_{k0}^{k_1}=\langle p_{k_1}(\text{cos}{\theta }_t)p_k(\text{cos}{\theta }_r)\rangle$$

(10)

The differential cross-section is given by:

$$\frac{1}{\sigma }\frac{d{\sigma }_{00}}{d{\omega }_t}\equiv p({\omega }_t)=\frac{1}{4\pi }\sum _{k_1}[k_1]h_0^{k_1}(k_1,0)p_{k_1} (\text{cos}{\theta }_t)$$

(11)

the bipolar moments $h_0^{k_1}(k_1,0)$ are evaluated using the expectation values of the Legendre moments of the differential cross-section: $S_{00}^{k_1}=h_0^{k_1}(k_1,0)=\langle p_k(\text{cos}{\theta }_t)\rangle$.

In many photoinitiated bimolecular reaction experiments, we will be sensitive to only those polarization moments with k = 0 and k = 2. In order to compare calculations with experiments, (2π/σ)(dσ₀₀/dω_t), (2π/σ)(dσ₂₀/dω_t), (2π/σ)(dσ₂₂₊/dω_t) and (2π/σ)(dσ₂₁/dω_t) are calculated. In the computation, PDDCSs are expanded up to k₁=7, which is sufficient for good convergence.

The usual two vector correlations (k - k’, k - j’, k’ - j’) are expanded in a series of Legendre polynomials. The distribution of the k - j’ correlation is characterized by P(θ_r) and the P(θ_r) can be written as [26–28]:

$$p({\theta }_r)=\frac{1}{2}\sum _k[k]a_0^k{p}_k(\text{cos} {\theta }_r)$$

(12)

where the $a_0^k$ coefficients (polarization parameters) are given by $a_0^k=\langle P_k(\text{cos}{\theta }_r)\rangle$ with the angular brackets stand for an average over all the reactive trajectories. In this paper, p(θ_r) is expended up to k=18, which shows good convergence.

The dihedral angle distribution of the k - k’ - j’ three-vector-correlation is characterized by angle φ_r [28,30]. It has been shown that the distribution of dihedral angle φ_r may be expanded as a Fourier series:

$$p({\phi }_r)=\frac{1}{2\pi }[1+\sum _{neven \ge 2}{a}_n \text{cos}(n{\phi }_r\text{)}+ \sum _{nodd \ge 1}{b}_n \text{sin}(n{\phi }_r)]$$

(13)

with a_n = 2〈cos(nφ_r)〉 and b_n = 〈2 sin(nφ_r)〉. In this computation, p(φ_r) is expanded to n=24, which shows good convergence.

The joint probability density function of angles θ_r and φ_r, which define the direction of j’, can be written [31] as:

$$p({\theta }_r,{\phi }_r)=\frac{1}{4\pi }\sum _{kq}[k]a_q^k{C}_{kq}{({\theta }_r,{\phi }_r)}^{*}=\frac{1}{4\pi }\sum _k\sum _{q\ge 0}[a_{q\pm }^k\text{cos}(q{\phi }_r)-a_{q\mp }^{k}i\text{sin}(q{\phi }_r)]C_{kq}({\theta }_r,0)$$

(14)

The polarization parameter $a_q^k$ is evaluated as:

$$\begin{array}{cc}{a}_{q\pm }^k=2\langle C_{k\left|q\right|}({\theta }_r,0)\text{cos}(q{\phi }_r)\rangle & \text{k} \text{is} \text{even},\end{array}$$

(15)

$$\begin{array}{cc}{a}_{q\mp }^k=2i\langle C_{k\left|q\right|}({\theta }_r,0)\text{sin}(q{\phi }_r)\rangle & \text{k} \text{is} \text{odd}\text{.}\end{array}$$

(16)

In the calculation, p(θ_r,φ_r) is expanded up to k=7, which is sufficient for good convergence.

3.2. Potential energy surface

The extended-London-Eyring-Polanyi-Sato (LEPS) potential energy surface (PES) is applied in our calculation [32]:

$$U(R_1,R_2,R_3,)=Q_1+Q_2+Q_3-{\left\{\frac{1}{2}[(J_1-J_2)+(J_2-J_3)+(J_3-J_1)]\right\}}^{\stackrel{1}{}/\underset{2}{}}$$

(17)

where Q_i =(¹E_i +³E_i)/2, J_i = (¹E_i −³E_i)/2. ¹E_i is defined as the diatomic Morse potential function, and ³E_i stands for the anti-Morse function:

$${}^1{E}_i={}^1{D}_i\left({\left\{1-\text{exp}[-{\beta }_i(\gamma -{\gamma }_0)]\right\}}^2-1\right),$$

(18)

$${}^3{E}_i={}^3{D}_i\left({\left\{1+\text{exp}[-{\beta }_i(\gamma -{\gamma }_0)]\right\}}^2-1\right).$$

(19)

where ³D_i = ¹D_i(1 - S_i)/2(1 + S_i) and S_i is an adjustable parameter. For the reaction of A + BC → AB + C, the subscript i = 1, 2, 3 indicate AB, BC and CA, respectively.

The experimental results and the ab initio calculations of the O + CH₄ → OH + CH₃ reaction show that the minimal barrier occurs at a collinear configuration. The alkyl radical does not possess significant internal excitation. So in this paper, the CH₄ is treated as a H-CH₃ pseudo-diatom for approximation. The classical Hamilton equations are integrated numerically for motion in three dimensions. Trajectories are initiated with the CH₄ molecule in v=0 and j=0 levels and the collision energy is 0.65 eV. 100,000 trajectories are sampled, and the integration step size in the trajectories is chosen to be 0.1 fs for the stability of the calculation results. The parameters of extended-LEPS PESs are presented in

Table 1. The PES parameters for the O + CH₄ → OH + CH₃ reaction.

Parameter	O-H	H-CH3	O-CH3
βe (Å−1)a	2.294	1.83	1.96
De(kJ/mol)a	445.34	447.26	384.35
rea	0.9706	1.093	1.44
Sato Ca	0.30	0.20	−0.15
Sato Db	0.70	0.3702	−0.4

^a Taken from Reference 33.

^b This work

[33]. All the calculations performed here were done under adiabatic approximation.

4. Conclusions

We have presented a quasiclassical trajectory study of the product polarization from the O + CH₄ → OH + CH₃ reaction on the new LEPS PES by using a new Sato parameter. The differential cross section (2π/σ)(dσ_kq/dω_t) show that the scattering is predominantly in the backward hemisphere. For the vector correlation, the J_OH is aligned in the plane perpendicular to the line of center and the four PDDCSs give a good explanation about the vector correlation. From the isotope effect for the reactions O + CH₄ and O + CD₄ we find that there is a backward scattering tendency with the deuterium instead of hydrogen of the (2π/σ)(dσ₀₀/dω_t) distributions. And the angular momentum polarization (P(θ_r) and P(φ_r)) of OD is much stronger than that of OH.