Effects of reagent rotational excitation on the H + CHD3 → H2 + CD3 reaction: A seven dimensional time-dependent wave packet study Zhaojun Zhang and Dong H. Zhang Citation: The Journal of Chemical Physics 141, 144309 (2014); doi: 10.1063/1.4897308 View online: http://dx.doi.org/10.1063/1.4897308 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Time-dependent quantum wave packet study of the Ar+H2 +→ArH++H reaction on a new ab initio potential energy surface for the ground electronic state (12 A′) J. Chem. Phys. 138, 174305 (2013); 10.1063/1.4803116 Effects of reagent vibrational excitation on the dynamics of the H + CHD3 → H2 + CD3 reaction: A sevendimensional time-dependent wave packet study J. Chem. Phys. 135, 024313 (2011); 10.1063/1.3609923 Evidence for excited spin-orbit state reaction dynamics in F + H 2 : Theory and experiment J. Chem. Phys. 128, 084313 (2008); 10.1063/1.2831412 Influence of rotation and isotope effects on the dynamics of the N ( D 2 ) + H 2 reactive system and of its deuterated variants J. Chem. Phys. 123, 224301 (2005); 10.1063/1.2131075 A time-dependent wave-packet quantum scattering study of the reaction H 2 + ( v = 0 – 2 , 4 , 6 ; j = 1 ) + He → HeH + + H J. Chem. Phys. 122, 244322 (2005); 10.1063/1.1948380
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THE JOURNAL OF CHEMICAL PHYSICS 141, 144309 (2014)
Effects of reagent rotational excitation on the H + CHD3 → H2 + CD3 reaction: A seven dimensional time-dependent wave packet study Zhaojun Zhang and Dong H. Zhanga) State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China
(Received 1 August 2014; accepted 25 September 2014; published online 10 October 2014) Seven-dimensional time-dependent wave packet calculations have been carried out for the title reaction to obtain reaction probabilities and cross sections for CHD3 in J0 = 1, 2 rotationally excited initial states with k0 = 0 − J0 (the projection of CHD3 rotational angular momentum on its C3 axis). Under the centrifugal sudden (CS) approximation, the initial states with the projection of the total angular momentum on the body fixed axis (K0 ) equal to k0 are found to be much more reactive, indicating strong dependence of reactivity on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . However, at the coupled-channel (CC) level this dependence becomes much weak although in general the K0 specified cross sections for the K0 = k0 initial states remain primary to the overall cross sections, implying the Coriolis coupling is important to the dynamics of the reaction. The calculated CS and CC integral cross sections obtained after K0 averaging for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, meaning that the initial rotational excitation of CHD3 up to J0 = 2, regardless of its initial k0 , does not have any effect on the total cross sections for the title reaction, and the errors introduced by the CS approximation on integral cross sections for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897308] I. INTRODUCTION
The H + CH4 → H2 + CH3 and its reverse reactions play important roles in CH4 /O2 combustion chemistry, and have therefore been the subject of both experiment and theoretical investigations.1–11, 15–36 Because five of the six atoms involved are hydrogens, it is an ideal candidate for high quality ab initio quantum chemistry calculations of the potential energy surface and quantum dynamics studies. Substantial efforts have been devoted on developing an accurate global potential energy surface (PES) for the system on which reliable dynamical calculations can be carried out.5, 15, 21, 22, 30, 36–39, 41 Recent years we have witnessed the constructions on PESs for the system at a quantitative level of accuracy by using various fitting methods based on high level ab initio calculations.42–48 During the past two decades, many dynamics methods have been applied to study the reaction system, with the number of degree of freedom included varying from 3 up to 12 (full-dimension). The reaction has become a benchmark for developing and testing theoretical methods for studying polyatomic chemical reactions. Takayanagi carried out the first quantum dynamical study of the reaction on Jordan-Gilbert PES. In that three-dimensional calculation, the system was treated as a collinear four-atom reaction model with CH3 group treated as a pseudodiatomic molecule.49 Yu and Nyman included the umbrella motion of CH3 in their four-dimensional rotating bond approximaa) Electronic mail: [email protected]
0021-9606/2014/141(14)/144309/8/$30.00
tion (RBA) treatment.4 The rate constants obtained from these two studies agree to each other quite well, but are much larger than the experimental results. Wang and Bowman carried out a six-dimensional (6D) time-dependent wave packet study by treating the three hydrogen atoms in CH3 as a pseudoatom.6 Wang and Zhang applied their fourdimensional semirigid vibrating rotor target (SVRT) model to the system.5 Later on, they extended their four-dimensional SVRT model by including the umbrella motion of CH3 group.8 The calculated five-dimensional (5D) SVRT rate constants were quite in good agreement with experiment, but significantly below the ones obtained from the previous threeand four-degree-of-freedom reduced dimensionality quantum calculations. Manthe and co-workers have performed full dimensional quantum dynamical calculations on the rate constants of this reaction based on the multiconfiguration time-dependent Hartree (MCTDH) method.12–15, 23, 39, 50, 51 Very recently, Welsch and Manthe successfully carried out full-dimensional quantum dynamics calculations on the H + CH4 → H2 + CH3 reaction at the state-to-state level based on the MCTDH approach.40 This group has been performing time-dependent wave packet studies of the system employing an eight-dimensional (8D) model.9, 22, 24, 36 The model was originally proposed by Palma and Clary by restricting the nonreacting CH3 group in a C3v symmetry.52 Since the assumption holds very well in reality, the model has a quantitative level of accuracy. Yang, Zhang, and Lee successfully implemented the model by using the time-dependent initial state selected wave packet method
141, 144309-1
© 2014 AIP Publishing LLC
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(ISSWP), and carried out seven-dimensional (7D) quantum dynamics studies for this reaction which the CH bond lengths in the nonreacting CH3 group were fixed.9 Later Zhang and Lu et al. carried out a transition state wave packet study for the H + CH4 reaction.24 In 2010, Zhang and Zhou et al. used the 7D model to study the H + CD4 → HD + CD3 reaction which showed the theoretical integral cross sections (ICS) has a good agreement with the experimental result.30 Zhou, Wang, and Zhang also investigated the effects of the CHD3 vibrational excitation for the H + CHD3 → H2 + CD3 using the 7D model which found that the C–H stretching excitation can promote the reaction dramatically.31 Recently, Liu et al. carried out a six-dimensional state-to-state quantum dynamics study of the H + CH4 → H2 + CH3 reaction which based on the 7D model by assuming that the CH3 group can rotate freely with respect to its C3 symmetry axis.33 In addition, a new eight-dimensional quantum mechanical Hamiltonian based on Palma and Clary 8D model for the X + YCZ3 → XY + CZ3 was proposed by Liu, Xiong, and Yang and has been used to study the H + CH4 reaction.32 Very recently, we carried out the first coupled channel (CC) calculation for the H + CHD3 → H2 + CD3 reaction using the 7D model,36 and found that the centrifugal sudden (CS) approximation considerably underestimated the CC results. We note that the experimental result is thermally averaged over the initial rotation of methane, whereas the initial state selected quantum results are for initial nonrotating methane. In this paper, we extend the initial state selected wave packet method to calculate probabilities and cross sections for the title reaction for CHD3 rotationally excited initial state. Recently, Wang, Lin, and Liu reported a new experimental result about the vibrational enhancement factor in the Cl + CHD3 reaction which agreed reasonably well with the 7D quantum dynamics calculation.53 In their paper, they also raised a question about how the initial rotational selection influences the ground and the vibrational excited reactivity. Our calculations may offer some clues for that question although we do not study that reaction directly. The content of this paper is arranged as follows: Section II outlines the method used in current time-dependent wave packet calculation. Results and discussions are showed in Sec. III. We conclude in Sec. IV. II. THEORY
The reactant Jacobi coordinates (R, r, χ , θ 1 , ϕ 1 , θ 2 , ϕ 2 ) for X + YCZ3 is shown in Fig. 1, with X, Y, Z referring to the incoming H atoms, the H atom in the CHD3 , the D atom, respectively, in this study. The time-dependent quantum
wave packet method used in the present study is an extension of earlier time-dependent wave packet studies of X + YCZ3 reaction.22, 36 The 7D Hamiltonian is written as ˆl2 ˆ 2 ¯2 ∂ 2 ¯2 ∂ 2 (Jˆ tot − J) Hˆ = − + − + + 2μR ∂R 2 2μr ∂r 2 2μR R 2 2μr r 2 ˆ vib ˆ rot +K CZ + KCZ + V (R, r, χ , θ1 , ϕ1 , θ2 , ϕ2 ),
(1)
where μR is reduced mass of X and YCZ3 , μr is reduced mass of Y and CZ3 , R is the distance from the center of mass of YCZ3 to X; r is the distance from the center of mass of CZ3 to Y; χ is the angle between a CZ bond and the C3 symmetry axis of CZ3 . The first two terms are the kinetic energy operators for R and r, respectively, Jˆ tot is the total angular momentum operator of the system, Jˆ is the rotational angular momentum of YCZ3 , and ˆl is the orbital angular momentum operator vib rot and KCZ are the vibrational of Y with respect to CZ3 . KCZ and rotational kinetic energy operators of CZ3 , respectively (no vibration-rotation coupling exists due to the symmetry requirement and the definition of the CZ3 -fixed frame), defined as 2 2 2 cos ∂2 χ χ ¯ sin vib Kˆ CZ = − 2 + 2s μx μs ∂χ 2 ∂ ¯2 1 1 sin χ cos χ (2) − 2 − s μs μx ∂χ and 1 ˆ2 rot Kˆ CZ j + = 2IA
1 1 − 2IC 2IA
jˆz2 ,
(3)
where μx and μs are related to the mass of atoms C and Z, μx = 3mz and μs = 3mc mz /(mc + 3mz ). IA and IC are the rotational inertia of CZ3 , defined as 3 2mc IA = mz s 2 sin2 χ + cos2 χ (4) 2 mc + 3mz and Ic = 3mz s 2 sin2 χ ,
(5)
where jˆ2 is the rotational angular momentum of CZ3 and jˆz2 is its z-component. The time-dependent wave function is expended in terms of the parity-adapted rotational basis function as J MKε n Cnntot n J lj k (t)Gnr (R)Fn (r)Hn Jtot Mε = r χ
r
χ
n,nr ,nχ KJ lk J MKε
× (χ )Jtotlj k
ˆ rˆ , sˆ ), (R,
(6)
J MKε
X χ s Z
R θ
2
C
r
ϕ1
θ
1
ϕ2
Y
FIG. 1. The eight-dimensional Jacobi coordinates for the X + YCZ3 system.
where Cnntot n J lj k (t) are time-dependent coefficients, n, nr , and r χ nχ are labels for the basis functions in R, r, and χ , respecn tively. Gnr are sine basis function for R which are dependent on nr for their spatial ranges to separate interaction region from asymptotic region. The basis functions Fn (r) and r Hn (χ ) are obtained by solving one-dimensional reference χ Hamiltonians for r and χ , respectively, hr (r) = −
1 ∂2 ref + Vr (r) 2μr ∂r 2
(7)
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J. Chem. Phys. 141, 144309 (2014)
and hχ (χ ) = ref
vib KCZ
+
ref Vχ (χ ),
(8)
2J0 + 1 rotational states with J0 states of them degenerate. As a result, we only need to calculate J0 + 1 initial rotational states for a given J0 quantum state.
ref
where Vr (r) and Vχ (χ ) are the corresponding reference potential. ¯ J M Kε
The body-fixed total angular momentum basis Jtotlj k in Eq. (6) are defined as Jtot J K ¯ 1 Jtot M Kε D¯ Y (ˆr , sˆ ) = J lj k 2(1 + δK δk ) MK j lk 0
0
Jtot −K YjJl−k (ˆr , sˆ ) , (9) +ε(−1)Jtot +J +l+j +k D¯ M−K J
tot where is the parity of the system and D¯ MK is defined as
Jtot ˆ = 2Jtot + 1 D ∗Jtot (α, β, γ ) D¯ MK (R) (10) MK 8π 2 with M and K being the projection of total angular momentum Jˆtot on the z axis of the space-fixed and body-fixed Jtot ˆ the Wigner rotation matrix, frames, respectively. D¯ MK (R), depends on Euler angles which rotate the space-fixed frame 2 . onto the body-fixed frame and are the eigenfunctions of Jˆtot JK The spherical harmonics Yj lk (ˆr , sˆ ) are given by
2l + 1 j JK J ¯ j m l 0|J mD¯ mk (ˆs ), DKm (ˆr ) Yj lk (ˆr , sˆ ) = 2J + 1 m (11) J (ˆr ) depends on Euler angles which rotate the where D¯ Km XYCZ3 body-fixed frame onto the YCZ3 -fixed frame, and j D¯ mk (ˆs ) depends on Euler angles which rotate the YCZ3 -fixed frame onto the CZ3 -fixed frame,
2J + 1 ∗J J ¯ DKm (0, θ1 , ϕ1 ), (12) DKm (ˆr ) = 4π
j D¯ mk (ˆs )
=
2j + 1 ∗j Dmk (0, θ2 , ϕ2 ). 4π
(13)
As in Refs. 36 and 54, we construct wave packet and propagate them to calculate the reaction probabilities J ε Pv totJ k K (E). The integral cross section from a specific initial 0 0 0 0 state (v0 J0 k0 ) is obtained by summing the reaction probabilities over all the partial waves (total angular momentum Jtot ) σv J k (E) 0 0 0
⎫ ⎧ ⎬ ⎨ π 1 Jtot
= (2J + 1)P (E) tot v0 j0 k0 K0 ⎭ (2J0 + 1) K ⎩ kE2 J ≥K 0
0
tot
(14) K
1 = σ 0 (E), (2J0 + 1) K v0 J0 k0
(15)
0
K ε where kE = 2μR E, and σv J0 k (E) is defined as K0 and 0 0 0 ε specified cross section. The CHD3 is a symmetric top molecule. For every rotational quantum number J0 , there are
III. RESULTS AND DISCUSSION
The time-dependent wave packet calculation was carried out on the Xu-Chen-Zhang (XCZ) potential energy surface, which was constructed recently by this group using neural network fittings based on 48 000 ab initio energy points calculated at UCCSD(T)-F12/AVTZ level. With a root mean square fitting error of 3.0 meV for geometries with ab initio energies less than 1.36 eV respect to H + CH4 , and 5.1 meV for energy between 1.36 eV and 2.72 eV, the PES is more accurate than all preceding PESs for the system according to the results of many tests as reported in Ref. 48. An L-shaped wave function expansion for R and r was used to reduce the size of the basis set.55 A total number of 100 sine basis functions covering a range from 3.0 to 15.0 bohrs were used for R with 30 grid points in the interaction region. For the CH bond(r), 6 and 40 vibrational basis functions in the range of [1.0,5.0]a0 were used for asymptotic and interaction region, respectively. The length of the CD bond was fixed at its equilibrium value of 2.06 bohrs, and 8 basis functions were used for umbrella motion. The size of rotational basis functions is controlled by the parameters Jmax = 56, lmax = 35, jmax = 21, and kmax = 3. Under the CS approximation we only need to include one K-block rotational basis functions in our calculations. For calculations without the CS approximation, we denote them as nK calculations with n referring to the number of K-blocks included in the calculations. In our present calculation, the number of K-blocks is up to 5 to obtain the converged results.36 We denote them as the CC results in contrast to the CS results. The size of rotational basis functions was up to 429 027. The center of the initial Gaussian wave packet was located at R0 = 13 bohrs with the width δ = 0.25 bohr, and the central energy was E0 = 0.85 eV. Table I lists the rotational excited energies of the quantum number of J0 = 1 and J0 = 2, and k0 > 0 states are doubly degenerate. Figure 2 shows the total reaction probabilities for the CHD3 initial rotational state (J0 ,k0 ) = (1,0) under the CS approximation, together with the reaction probabilities for the ground rovibrational initial state for Jtot = 0. As seen from Fig. 2(a) that the reaction probabilities for the K0 = 0 initial state rise quickly from the threshold energy regions, reach the maximum values and then begin to decline slowly with the further increase of collision energy. In contrast, the reaction TABLE I. The rotational eigenenergies (in cm−1 ) of the CHD3 measured from the ground state energy of 3341.315 cm−1 , and k0 are the projection of the rotational quantum number on the symmetry axis of the CHD3 . J0
k0
Energy
1 1 2 2 2
0 1 0 1 2
6.588 5.955 19.771 19.137 17.235
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144309-4
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
0.4
Jtot=0 Jtot=20
5.0
Jtot=40 J0=0 Jtot=0
(a)
Jtot=20 Jtot=40
0.2
(a)
Jtot=21 Jtot=41
(b) 0.8
K0=1
-3
Probability(10 )
Jtot=1
0 0.4
Jtot=0
0.0 K0=1
-3
Probability(10 )
0.0
0.2
K0=0
-3
Probability(10 )
K0=0
-3
Probability(10 )
10.0
1.2
1.6
Translational energy (eV) FIG. 2. (a) Total reaction probabilities for the CHD3 initial rotational state (J0 = 1, k0 = 0, K0 = 0) under the CS approximation and the reaction probabilities for the ground rovibrational initial state for Jtot = 0; (b) Total reaction probabilities for the CHD3 initial rotational state (J0 = 1, k0 = 0, K0 = 1) under the CS approximation.
probabilities for the K0 = 1 initial state monotonically in the whole energy region we studied as shown in Fig. 2(b). It is apparent that for the (1,0) initial state the reaction probabilities for K0 = 0 shown in Fig. 2(a) are substantially larger than the corresponding K0 = 1 probabilities shown in Fig. 2(b) in the whole energy region considered here. For Jtot = 0, the K0 = 0 probabilities for the (1,0) initial state are larger than the corresponding values for the J0 = 0 initial state as shown in Fig. 2(a) by about a factor of 3. It also can be seen from the figure that both for K0 = 0 and K0 = 1 states the reaction thresholds shift regularly with the increase of Jtot due to the centrifugal potential. In Fig. 3, we show the total reaction probabilities for the CHD3 initial rotational state (J0 ,k0 ) = (1,1) under the CS approximation. The behavior of the reaction probabilities in Figs. 3(a) and 3(b) is totally opposite to those for the (1,0) initial state with respect to the initial K0 values, with the probabilities for K0 = 1 in Fig. 3(b) substantially larger than the corresponding K0 = 0 probabilities in Fig. 3(a). In fact, the probabilities for the (J0 ,k0 ,K0 ) = (1,0,1) initial state in Fig. 2(b) are essentially the same as those for the (1,1,0) state shown in Fig. 3(a), and the probabilities for the (1,0,0) initial state in Fig. 2(a) are about twice as large as those for the (1,1,1) state in Fig. 3(b). The very different behaviors of the reaction probabilities for the (1,0) and (1,1) initial states, as well as for the different initial K0 can be understood as following. Since the CHD3 molecule is a symmetric top rotor with the CH bond as the symmetry axis, k0 is a good quantum number. For the initial (1,1) state, the rotational angular momentum has a projection of 1 on the CH axis, or the molecule rotates around the CH bond. In contrast, for the initial (1,0) state, the rotational angular momentum has a projection of 0 on the CH axis, or the molecule rotates around an axis perpendicular to the CH
Jtot=1
2.5
Jtot=21 Jtot=41
(b) 0.0 0.4
0.8
1.2
1.6
Translational energy (eV) FIG. 3. (a) Same as Fig. 2(a) except for (J0 = 1, k0 = 1, K0 = 0) state; (b) Same as Fig. 2(b) except for (J0 = 1, k0 = 1, K0 = 1) state.
bond. When the incoming H atom collides with the CHD3 molecule, because the projection of the orbital momentum on the body-fixed axis is equal to zero, the initial K0 of the system is given by the projection of CHD3 rotational angular momentum on the body-fixed axis. For the initial (J0 ,k0 ,K0 ) = (1,0,0) and (1,1,1) states, the initial projection of total angular momentum on body-fixed axis K0 equal to k0 means the body-fixed axis is parallel to the CH bond, or the incoming H atom approaches along the CH bond. As a result, these initial states have large reaction probabilities because the reaction has a collinear saddle geometry. In contrast, for the initial (J0 ,k0 ,K0 ) = (1,0,1) and (1,1,0) states, the body-fixed axis is perpendicular to the CH bond, or the incoming H atom approaches perpendicular to the CH bond, resulting in much smaller reaction probabilities as compared to the (1,0,0) and (1,1,1) states. In Ref. 36, we calculated the CC integral cross sections for the title reaction by including sufficient numbers of Kblocks for Jtot > 0, and found the CC ICSs are considerably larger than the corresponding CS results, in particular for the ground vibrational state. To investigate the accuracy of the CS approximation for the rotationally excited state, we carried out CC calculations for these rotationally excited initial states. Figure 4 shows the CC total reaction probabilities for the initial state (J0 = 1,k0 = 0). For the K0 = 0 initial state shown in Fig. 4(a), the CC reaction probabilities shift regularly with the increasing of Jtot , slightly slower than the CS results shown in Fig. 2(a), however the CC probabilities for Jtot = 20, 40, in particular for Jtot = 40 are considerably smaller than the CS results in high energy region. Figure 4(c) shows the reaction probabilities for a few Jtot with K0 = 1 and ε = −1 (denoted as K0 = 1− ). As seen, the CC probabilities are slightly larger than the corresponding CS ones, and the thresholds for CC probabilities are also slightly lower. In strong contrast, the CC probabilities for K0 = 1+ shown in Fig. 4(b) behave very differently from the corresponding CS ones in Fig. 2(b), as well
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J. Chem. Phys. 141, 144309 (2014)
Probability(10 )
4.0
5.0
Jtot=0 Jtot=20 Jtot=40
(a)
Jtot=20 Jtot=40
2.0
(a)
Jtot=41
4.0
+
K0=1
-3
Probability(10 )
Jtot=1 Jtot=21
(b)
0.0
Jtot=1 Jtot=21
2.5
Jtot=41
(b)
0.0 Jtot=1 Jtot=21
0.25
Jtot=41
(c)
0 0.4
0.8
-
K0=1
-3
Probability(10 )
-
K0=1
-3
Jtot=0
0.0 +
K0=1
-3
Probability(10 )
0.0
Probability(10 )
K0=0
-3
K0=0
-3
Probability(10 )
10.0
1.2
Jtot=1
2.5
Jtot=21 Jtot=41
(c)
0.0 0.4
1.6
Translational energy (eV)
0.8
1.2
1.6
Translational energy (eV)
FIG. 4. (a) CC total reaction probabilities for the initial state (J0 = 1, k0 = 0, K0 = 0); (b) Same as Fig. 4(a) except for (J0 = 1, k0 = 0, K0 = 1+ ) state; (c) Same as Fig. 4(a) except for (J0 = 1, k0 = 0, K0 = 1− ) state.
FIG. 5. (a) Same as Fig. 4(a) except for (J0 = 1, k0 = 1, K0 = 0) state; (b) Same as Fig. 4(b) except for (J0 = 1, k0 = 1, K0 = 1+ ) state; (c) Same as Fig. 4(c) except for (J0 = 1, k0 = 1, K0 = 1− ) state.
as its K0 = 1− counterparts shown in Fig. 4(c). And the difference increases with the increase of Jtot s. For the first few Jtot , the threshold energies actually decrease with the increase of Jtot . More importantly, the level-off probability values significantly increase with the increase of Jtot . Obviously, the Coriolis coupling between different K blocks in the CC calculations substantially change the reaction probabilities for this initial state, in particular for the K0 = 1+ case. For initial K0 = 0 shown in Fig. 4(a), coupling to the K > 0 blocks reduces the total reaction probabilities, in particular when Jtot is large, because the K = 0 state is much more reactive for the (J0 = 1,k0 = 0) initial state as shown in Fig. 2. On the other hand, for initial K0 = 1+ , the coupling to the K = 0 block substantially enhances reactivity as shown in Fig. 4(b). For initial K0 = 1− , because the rotational basis functions with K = 0 do not exist for k = 0 due to parity constrain as can be seen from Eq. (9), there is no coupling to K = 0 block until k is excited from initial 0 to 3, or larger. As a result, the CC probabilities for K0 = 1− are only slightly larger than the corresponding CS ones. Figure 5 shows the CC total reaction probabilities for the initial state (J0 = 1,k0 = 1). From Fig. 5(a) we can see that the behavior of K0 = 0 initial state is very similar to that for (J0 = 1,k0 = 0, K0 = 1+ ) as shown in Fig. 4(b). Apparently, the coupling to the K = 1 block substantially enhances the reactivity for large Jtot because for this initial state the K = 1 state is much more reactive as shown in Fig. 3. While for the K0 = 1 case, the coupling to the K = 0 block slightly reduces the probabilities for both parities as shown in Figs. 5(b) and 5(c), compared to the corresponding CS results. In order to obtain the integral cross section, we calculated the total reaction probabilities for Jtot up to 65 to converge the ICS for collision energy up to 1.6 eV. In Fig. 6, we present the CS and CC K0 specified cross sections
for the initial state (J0 = 1,k0 = 0). The CS cross sections steadily increase with the increase of collision energy, with σ (K0 = 0) much larger than σ (K0 = 1), reaching 0.27 a02 at Ec = 1.6 eV. The CC σ (K0 = 0) is a little bit larger than the CS σ (K0 = 0) for collision energy up to 0.7 eV. It reaches the maximum value at around 1.1 eV, then begins to decline slowly with the further increase of collision energy. At Ec = 1.6 eV, the CC σ (K0 = 0) is 0.16 a02 , considerably smaller than the CS value. The CC σ (K0 = 1− ) is only slightly larger than the CS σ (K0 = 1), both are negligibly small as compared to their counterparts. In strong contrast, the CC σ (K0 = 1+ ) is much larger than the σ (K0 = 1− ), even larger than the CC σ (K0 = 0) when the collision energy is higher than 1.3 eV. Figure 7 presents the CS and CC K0 specified cross sections for the initial state (J0 = 1, k0 = 1). The CS σ (K0 = 1) is much larger than the σ (K0 = 0), totally opposite to the initial
CS K0=0
2
Cross Section (a0 )
0.25
CS K0=1 CC K0=0 +
0.2
CC K0=1
-
CC K0=1
0.15 0.1 0.05 0 0.4
0.8
1.2
1.6
Translational Energy (eV) FIG. 6. CS and CC K0 specified cross sections for the initial state (J0 = 1, k0 = 0).
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J. Chem. Phys. 141, 144309 (2014)
0.2
0.25
CC K0=1
0.1
CC K0=1
0.2
+ -
0.05
CS K0=2 CC K0=0
2
0.15
CS K0=1
Cross Section(a0 )
2
Cross Section (a0 )
CS K0=0
CS K0=0 CS K0=1 CC K0=0
+
CC K0=1
-
0.15
CC K0=1
+
CC K0=2
-
CC K0=2
0.1
0.05
0 0.4
0.8
1.2
0 0.4
1.6
0.8
1.2
1.6
Translational Energy (eV)
Translational Energy (eV) FIG. 7. Same as Fig. 6 except for (J0 = 1, k0 = 1) state.
FIG. 9. Same as Fig. 6 except for (J0 = 2, k0 = 1) state.
state (J0 =1, k0 = 0) shown in Fig. 6. In contrast, the differences between the CC cross sections for K0 = 0 and 1 are not big, in particular in the high energy region. The CC σ (K0 = 1) for both parities are quite close, in particular in the low energy region. The CC σ (K0 = 0) are considerably smaller than σ (K0 = 1) in the low energy, but surpass σ (K0 = 1− ) at Ec = 1.4 eV. The K0 specified CS cross sections in Figs. 6 and 7 show very strong steric effect of the reaction, i.e., the reactivity strongly dependent on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . The CC σ (K0 = 0) for the (J0 = 1, k0 = 0) initial state is still considerably larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) as can be seen from Fig. 6, in particular in the low collision energy region, therefore rather strong steric effect still persist for that state although as not strong as in the CS case. However, for the (J0 = 1, k0 = 1) initial state the CC σ (K0 = 0) is only slightly larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) as shown in Fig. 7, indicating steric effect is very weak for the state. We also carried out CS and CC calculations for the J0 = 2 initial states with k0 = 0, 1, 2. Due to a large number of reaction probabilities obtained, we only present K0 specified cross sections in Figs. 8–10. From Fig. 8, we can see that for
the (J0 = 2, k0 = 0) initial state the CS K0 specified CS cross sections show very strong steric effect, with σ (K0 = 0) much larger than σ (K0 = 1) and σ (K0 = 2) close to zero, similar to that for the (J0 = 1, k0 = 0) initial state shown in Fig. 6. At the CC level σ (K0 = 0) is larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) only in the low collision energy region, while σ (K0 = 2− ) remains negligible compared to its K0 = 0,1 counterparts. For the (J0 = 2, k0 = 1) initial state shown in Fig. 9, the CS σ (K0 = 1) are much larger than the other two. However, the CC cross sections behave very differently. For this initial state, σ (K0 = 1− ) and σ (K0 = 2− ) are larger than their positive parity counterparts. As can be seen from Fig. 10 the CS cross sections for the (J0 = 2, k0 = 2) initial state are very similar to those for the (J0 = 1,k0 = 1) with K0 = k0 state much more reactive. While at the CC level σ (K0 = 2) are larger than the other two only in the low energy region. Therefore, as for the J0 = 1 initial state, very strong steric effects for the J0 = 2 initial state only exist under the CS approximation. Finally, in Fig. 11 we show the CS and CC cross sections for the J0 = 0, 1, 2 initial rotational states. No matter at the CS or CC level, the K0 specified cross sections for the J0 = 1, 2 initial states manifest dramatic differences from the J0 = 0 initial state as shown in Figs. 6–10. However, after K0 0.25
0.4
CS K0=0
CS K0=0
CS K0=1
CS K0=1
Cross Section(a0 )
+
0.3
CC K0=1
-
CC K0=1
+
CC K0=2
-
0.2
CC K0=2
0.1
0 0.4
0.2
CS K0=2 CC K0=0
2
2
Cross Section(a0 )
CS K0=2 CC K0=0
+
CC K0=1
-
0.15
CC K0=1
+
CC K0=2
-
CC K0=2
0.1
0.05
0.8
1.2
Translational Energy (eV) FIG. 8. Same as Fig. 6 except for (J0 = 2, k0 = 0) state.
1.6
0 0.4
0.8
1.2
1.6
Translational Energy (eV) FIG. 10. Same as Fig. 6 except for (J0 = 2, k0 = 2) state.
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J. Chem. Phys. 141, 144309 (2014)
regardless of its initial k0 , does not have any effect on the total cross sections. It also means that the errors introduced by the CS approximation on ICS for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state.
2
Cross Section(a0 )
CC
0.1 CS
ACKNOWLEDGMENTS
(0,0) (1,0) (1,1) (2,0) (2,1) (2,2)
0.05
0 0.4
0.8
1.2
This work was supported by the National Natural Science Foundation of China (Grant No. 90921014), Ministry of Science and Technology of China (2013CB834601), and Chinese Academy of Sciences. 1.6
Translational Energy (eV) FIG. 11. CS and CC cross sections for the J0 = 0, 1, 2 initial rotational states.
averaging the obtained CS and CC cross sections for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, meaning that the rotational excitation of the initial CHD3 up to J0 = 2 does not have any effect on the total cross sections for the title reaction. It is also interesting to see that the errors introduced by the CS approximation on ICS for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state reported in Ref. 36. IV. CONCLUSIONS
The time-dependent wave packet method has been applied to calculate the CS and CC probabilities and cross sections for the title reaction with CHD3 in J0 = 1, 2 rotationally excited initial state under the C3v approximation on a new potential energy surface recently constructed by this group using neural network fitting.48 At the CS level, it is found that the K0 = k0 initial states are much more reactive, meaning that the reactivity strongly dependent on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . However, at the CC level this dependence becomes much weak although in general the K0 specified cross sections for the K0 = k0 initial states remains primary to the overall cross sections, indicating the Coriolis coupling between different K blocks are important to the dynamics of the title reaction. This is in agreement with what we found that the CC cross sections are considerably larger than the corresponding CS results due to the strong Coriolis coupling.36 More calculations should be carried out on the other X + YCZ3 type reactions to investigate if the failure of the CS approximation to accurately describe the dynamics is general to the complete X + YCZ3 family. Our calculations revealed that the overall CS and CC cross sections obtained after K0 averaging for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, despite the fact that the K0 specified cross sections for the J0 = 1, 2 initial states exhibit dramatic differences from the J0 = 0 initial state. This clearly showed that for the title reaction the initial rotational excitation of CHD3 up to J0 = 2,
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THE JOURNAL OF CHEMICAL PHYSICS 141, 144309 (2014)
Effects of reagent rotational excitation on the H + CHD3 → H2 + CD3 reaction: A seven dimensional time-dependent wave packet study Zhaojun Zhang and Dong H. Zhanga) State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China
(Received 1 August 2014; accepted 25 September 2014; published online 10 October 2014) Seven-dimensional time-dependent wave packet calculations have been carried out for the title reaction to obtain reaction probabilities and cross sections for CHD3 in J0 = 1, 2 rotationally excited initial states with k0 = 0 − J0 (the projection of CHD3 rotational angular momentum on its C3 axis). Under the centrifugal sudden (CS) approximation, the initial states with the projection of the total angular momentum on the body fixed axis (K0 ) equal to k0 are found to be much more reactive, indicating strong dependence of reactivity on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . However, at the coupled-channel (CC) level this dependence becomes much weak although in general the K0 specified cross sections for the K0 = k0 initial states remain primary to the overall cross sections, implying the Coriolis coupling is important to the dynamics of the reaction. The calculated CS and CC integral cross sections obtained after K0 averaging for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, meaning that the initial rotational excitation of CHD3 up to J0 = 2, regardless of its initial k0 , does not have any effect on the total cross sections for the title reaction, and the errors introduced by the CS approximation on integral cross sections for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897308] I. INTRODUCTION
The H + CH4 → H2 + CH3 and its reverse reactions play important roles in CH4 /O2 combustion chemistry, and have therefore been the subject of both experiment and theoretical investigations.1–11, 15–36 Because five of the six atoms involved are hydrogens, it is an ideal candidate for high quality ab initio quantum chemistry calculations of the potential energy surface and quantum dynamics studies. Substantial efforts have been devoted on developing an accurate global potential energy surface (PES) for the system on which reliable dynamical calculations can be carried out.5, 15, 21, 22, 30, 36–39, 41 Recent years we have witnessed the constructions on PESs for the system at a quantitative level of accuracy by using various fitting methods based on high level ab initio calculations.42–48 During the past two decades, many dynamics methods have been applied to study the reaction system, with the number of degree of freedom included varying from 3 up to 12 (full-dimension). The reaction has become a benchmark for developing and testing theoretical methods for studying polyatomic chemical reactions. Takayanagi carried out the first quantum dynamical study of the reaction on Jordan-Gilbert PES. In that three-dimensional calculation, the system was treated as a collinear four-atom reaction model with CH3 group treated as a pseudodiatomic molecule.49 Yu and Nyman included the umbrella motion of CH3 in their four-dimensional rotating bond approximaa) Electronic mail: [email protected]
0021-9606/2014/141(14)/144309/8/$30.00
tion (RBA) treatment.4 The rate constants obtained from these two studies agree to each other quite well, but are much larger than the experimental results. Wang and Bowman carried out a six-dimensional (6D) time-dependent wave packet study by treating the three hydrogen atoms in CH3 as a pseudoatom.6 Wang and Zhang applied their fourdimensional semirigid vibrating rotor target (SVRT) model to the system.5 Later on, they extended their four-dimensional SVRT model by including the umbrella motion of CH3 group.8 The calculated five-dimensional (5D) SVRT rate constants were quite in good agreement with experiment, but significantly below the ones obtained from the previous threeand four-degree-of-freedom reduced dimensionality quantum calculations. Manthe and co-workers have performed full dimensional quantum dynamical calculations on the rate constants of this reaction based on the multiconfiguration time-dependent Hartree (MCTDH) method.12–15, 23, 39, 50, 51 Very recently, Welsch and Manthe successfully carried out full-dimensional quantum dynamics calculations on the H + CH4 → H2 + CH3 reaction at the state-to-state level based on the MCTDH approach.40 This group has been performing time-dependent wave packet studies of the system employing an eight-dimensional (8D) model.9, 22, 24, 36 The model was originally proposed by Palma and Clary by restricting the nonreacting CH3 group in a C3v symmetry.52 Since the assumption holds very well in reality, the model has a quantitative level of accuracy. Yang, Zhang, and Lee successfully implemented the model by using the time-dependent initial state selected wave packet method
141, 144309-1
© 2014 AIP Publishing LLC
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144309-2
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
(ISSWP), and carried out seven-dimensional (7D) quantum dynamics studies for this reaction which the CH bond lengths in the nonreacting CH3 group were fixed.9 Later Zhang and Lu et al. carried out a transition state wave packet study for the H + CH4 reaction.24 In 2010, Zhang and Zhou et al. used the 7D model to study the H + CD4 → HD + CD3 reaction which showed the theoretical integral cross sections (ICS) has a good agreement with the experimental result.30 Zhou, Wang, and Zhang also investigated the effects of the CHD3 vibrational excitation for the H + CHD3 → H2 + CD3 using the 7D model which found that the C–H stretching excitation can promote the reaction dramatically.31 Recently, Liu et al. carried out a six-dimensional state-to-state quantum dynamics study of the H + CH4 → H2 + CH3 reaction which based on the 7D model by assuming that the CH3 group can rotate freely with respect to its C3 symmetry axis.33 In addition, a new eight-dimensional quantum mechanical Hamiltonian based on Palma and Clary 8D model for the X + YCZ3 → XY + CZ3 was proposed by Liu, Xiong, and Yang and has been used to study the H + CH4 reaction.32 Very recently, we carried out the first coupled channel (CC) calculation for the H + CHD3 → H2 + CD3 reaction using the 7D model,36 and found that the centrifugal sudden (CS) approximation considerably underestimated the CC results. We note that the experimental result is thermally averaged over the initial rotation of methane, whereas the initial state selected quantum results are for initial nonrotating methane. In this paper, we extend the initial state selected wave packet method to calculate probabilities and cross sections for the title reaction for CHD3 rotationally excited initial state. Recently, Wang, Lin, and Liu reported a new experimental result about the vibrational enhancement factor in the Cl + CHD3 reaction which agreed reasonably well with the 7D quantum dynamics calculation.53 In their paper, they also raised a question about how the initial rotational selection influences the ground and the vibrational excited reactivity. Our calculations may offer some clues for that question although we do not study that reaction directly. The content of this paper is arranged as follows: Section II outlines the method used in current time-dependent wave packet calculation. Results and discussions are showed in Sec. III. We conclude in Sec. IV. II. THEORY
The reactant Jacobi coordinates (R, r, χ , θ 1 , ϕ 1 , θ 2 , ϕ 2 ) for X + YCZ3 is shown in Fig. 1, with X, Y, Z referring to the incoming H atoms, the H atom in the CHD3 , the D atom, respectively, in this study. The time-dependent quantum
wave packet method used in the present study is an extension of earlier time-dependent wave packet studies of X + YCZ3 reaction.22, 36 The 7D Hamiltonian is written as ˆl2 ˆ 2 ¯2 ∂ 2 ¯2 ∂ 2 (Jˆ tot − J) Hˆ = − + − + + 2μR ∂R 2 2μr ∂r 2 2μR R 2 2μr r 2 ˆ vib ˆ rot +K CZ + KCZ + V (R, r, χ , θ1 , ϕ1 , θ2 , ϕ2 ),
(1)
where μR is reduced mass of X and YCZ3 , μr is reduced mass of Y and CZ3 , R is the distance from the center of mass of YCZ3 to X; r is the distance from the center of mass of CZ3 to Y; χ is the angle between a CZ bond and the C3 symmetry axis of CZ3 . The first two terms are the kinetic energy operators for R and r, respectively, Jˆ tot is the total angular momentum operator of the system, Jˆ is the rotational angular momentum of YCZ3 , and ˆl is the orbital angular momentum operator vib rot and KCZ are the vibrational of Y with respect to CZ3 . KCZ and rotational kinetic energy operators of CZ3 , respectively (no vibration-rotation coupling exists due to the symmetry requirement and the definition of the CZ3 -fixed frame), defined as 2 2 2 cos ∂2 χ χ ¯ sin vib Kˆ CZ = − 2 + 2s μx μs ∂χ 2 ∂ ¯2 1 1 sin χ cos χ (2) − 2 − s μs μx ∂χ and 1 ˆ2 rot Kˆ CZ j + = 2IA
1 1 − 2IC 2IA
jˆz2 ,
(3)
where μx and μs are related to the mass of atoms C and Z, μx = 3mz and μs = 3mc mz /(mc + 3mz ). IA and IC are the rotational inertia of CZ3 , defined as 3 2mc IA = mz s 2 sin2 χ + cos2 χ (4) 2 mc + 3mz and Ic = 3mz s 2 sin2 χ ,
(5)
where jˆ2 is the rotational angular momentum of CZ3 and jˆz2 is its z-component. The time-dependent wave function is expended in terms of the parity-adapted rotational basis function as J MKε n Cnntot n J lj k (t)Gnr (R)Fn (r)Hn Jtot Mε = r χ
r
χ
n,nr ,nχ KJ lk J MKε
× (χ )Jtotlj k
ˆ rˆ , sˆ ), (R,
(6)
J MKε
X χ s Z
R θ
2
C
r
ϕ1
θ
1
ϕ2
Y
FIG. 1. The eight-dimensional Jacobi coordinates for the X + YCZ3 system.
where Cnntot n J lj k (t) are time-dependent coefficients, n, nr , and r χ nχ are labels for the basis functions in R, r, and χ , respecn tively. Gnr are sine basis function for R which are dependent on nr for their spatial ranges to separate interaction region from asymptotic region. The basis functions Fn (r) and r Hn (χ ) are obtained by solving one-dimensional reference χ Hamiltonians for r and χ , respectively, hr (r) = −
1 ∂2 ref + Vr (r) 2μr ∂r 2
(7)
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J. Chem. Phys. 141, 144309 (2014)
and hχ (χ ) = ref
vib KCZ
+
ref Vχ (χ ),
(8)
2J0 + 1 rotational states with J0 states of them degenerate. As a result, we only need to calculate J0 + 1 initial rotational states for a given J0 quantum state.
ref
where Vr (r) and Vχ (χ ) are the corresponding reference potential. ¯ J M Kε
The body-fixed total angular momentum basis Jtotlj k in Eq. (6) are defined as Jtot J K ¯ 1 Jtot M Kε D¯ Y (ˆr , sˆ ) = J lj k 2(1 + δK δk ) MK j lk 0
0
Jtot −K YjJl−k (ˆr , sˆ ) , (9) +ε(−1)Jtot +J +l+j +k D¯ M−K J
tot where is the parity of the system and D¯ MK is defined as
Jtot ˆ = 2Jtot + 1 D ∗Jtot (α, β, γ ) D¯ MK (R) (10) MK 8π 2 with M and K being the projection of total angular momentum Jˆtot on the z axis of the space-fixed and body-fixed Jtot ˆ the Wigner rotation matrix, frames, respectively. D¯ MK (R), depends on Euler angles which rotate the space-fixed frame 2 . onto the body-fixed frame and are the eigenfunctions of Jˆtot JK The spherical harmonics Yj lk (ˆr , sˆ ) are given by
2l + 1 j JK J ¯ j m l 0|J mD¯ mk (ˆs ), DKm (ˆr ) Yj lk (ˆr , sˆ ) = 2J + 1 m (11) J (ˆr ) depends on Euler angles which rotate the where D¯ Km XYCZ3 body-fixed frame onto the YCZ3 -fixed frame, and j D¯ mk (ˆs ) depends on Euler angles which rotate the YCZ3 -fixed frame onto the CZ3 -fixed frame,
2J + 1 ∗J J ¯ DKm (0, θ1 , ϕ1 ), (12) DKm (ˆr ) = 4π
j D¯ mk (ˆs )
=
2j + 1 ∗j Dmk (0, θ2 , ϕ2 ). 4π
(13)
As in Refs. 36 and 54, we construct wave packet and propagate them to calculate the reaction probabilities J ε Pv totJ k K (E). The integral cross section from a specific initial 0 0 0 0 state (v0 J0 k0 ) is obtained by summing the reaction probabilities over all the partial waves (total angular momentum Jtot ) σv J k (E) 0 0 0
⎫ ⎧ ⎬ ⎨ π 1 Jtot
= (2J + 1)P (E) tot v0 j0 k0 K0 ⎭ (2J0 + 1) K ⎩ kE2 J ≥K 0
0
tot
(14) K
1 = σ 0 (E), (2J0 + 1) K v0 J0 k0
(15)
0
K ε where kE = 2μR E, and σv J0 k (E) is defined as K0 and 0 0 0 ε specified cross section. The CHD3 is a symmetric top molecule. For every rotational quantum number J0 , there are
III. RESULTS AND DISCUSSION
The time-dependent wave packet calculation was carried out on the Xu-Chen-Zhang (XCZ) potential energy surface, which was constructed recently by this group using neural network fittings based on 48 000 ab initio energy points calculated at UCCSD(T)-F12/AVTZ level. With a root mean square fitting error of 3.0 meV for geometries with ab initio energies less than 1.36 eV respect to H + CH4 , and 5.1 meV for energy between 1.36 eV and 2.72 eV, the PES is more accurate than all preceding PESs for the system according to the results of many tests as reported in Ref. 48. An L-shaped wave function expansion for R and r was used to reduce the size of the basis set.55 A total number of 100 sine basis functions covering a range from 3.0 to 15.0 bohrs were used for R with 30 grid points in the interaction region. For the CH bond(r), 6 and 40 vibrational basis functions in the range of [1.0,5.0]a0 were used for asymptotic and interaction region, respectively. The length of the CD bond was fixed at its equilibrium value of 2.06 bohrs, and 8 basis functions were used for umbrella motion. The size of rotational basis functions is controlled by the parameters Jmax = 56, lmax = 35, jmax = 21, and kmax = 3. Under the CS approximation we only need to include one K-block rotational basis functions in our calculations. For calculations without the CS approximation, we denote them as nK calculations with n referring to the number of K-blocks included in the calculations. In our present calculation, the number of K-blocks is up to 5 to obtain the converged results.36 We denote them as the CC results in contrast to the CS results. The size of rotational basis functions was up to 429 027. The center of the initial Gaussian wave packet was located at R0 = 13 bohrs with the width δ = 0.25 bohr, and the central energy was E0 = 0.85 eV. Table I lists the rotational excited energies of the quantum number of J0 = 1 and J0 = 2, and k0 > 0 states are doubly degenerate. Figure 2 shows the total reaction probabilities for the CHD3 initial rotational state (J0 ,k0 ) = (1,0) under the CS approximation, together with the reaction probabilities for the ground rovibrational initial state for Jtot = 0. As seen from Fig. 2(a) that the reaction probabilities for the K0 = 0 initial state rise quickly from the threshold energy regions, reach the maximum values and then begin to decline slowly with the further increase of collision energy. In contrast, the reaction TABLE I. The rotational eigenenergies (in cm−1 ) of the CHD3 measured from the ground state energy of 3341.315 cm−1 , and k0 are the projection of the rotational quantum number on the symmetry axis of the CHD3 . J0
k0
Energy
1 1 2 2 2
0 1 0 1 2
6.588 5.955 19.771 19.137 17.235
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144309-4
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
0.4
Jtot=0 Jtot=20
5.0
Jtot=40 J0=0 Jtot=0
(a)
Jtot=20 Jtot=40
0.2
(a)
Jtot=21 Jtot=41
(b) 0.8
K0=1
-3
Probability(10 )
Jtot=1
0 0.4
Jtot=0
0.0 K0=1
-3
Probability(10 )
0.0
0.2
K0=0
-3
Probability(10 )
K0=0
-3
Probability(10 )
10.0
1.2
1.6
Translational energy (eV) FIG. 2. (a) Total reaction probabilities for the CHD3 initial rotational state (J0 = 1, k0 = 0, K0 = 0) under the CS approximation and the reaction probabilities for the ground rovibrational initial state for Jtot = 0; (b) Total reaction probabilities for the CHD3 initial rotational state (J0 = 1, k0 = 0, K0 = 1) under the CS approximation.
probabilities for the K0 = 1 initial state monotonically in the whole energy region we studied as shown in Fig. 2(b). It is apparent that for the (1,0) initial state the reaction probabilities for K0 = 0 shown in Fig. 2(a) are substantially larger than the corresponding K0 = 1 probabilities shown in Fig. 2(b) in the whole energy region considered here. For Jtot = 0, the K0 = 0 probabilities for the (1,0) initial state are larger than the corresponding values for the J0 = 0 initial state as shown in Fig. 2(a) by about a factor of 3. It also can be seen from the figure that both for K0 = 0 and K0 = 1 states the reaction thresholds shift regularly with the increase of Jtot due to the centrifugal potential. In Fig. 3, we show the total reaction probabilities for the CHD3 initial rotational state (J0 ,k0 ) = (1,1) under the CS approximation. The behavior of the reaction probabilities in Figs. 3(a) and 3(b) is totally opposite to those for the (1,0) initial state with respect to the initial K0 values, with the probabilities for K0 = 1 in Fig. 3(b) substantially larger than the corresponding K0 = 0 probabilities in Fig. 3(a). In fact, the probabilities for the (J0 ,k0 ,K0 ) = (1,0,1) initial state in Fig. 2(b) are essentially the same as those for the (1,1,0) state shown in Fig. 3(a), and the probabilities for the (1,0,0) initial state in Fig. 2(a) are about twice as large as those for the (1,1,1) state in Fig. 3(b). The very different behaviors of the reaction probabilities for the (1,0) and (1,1) initial states, as well as for the different initial K0 can be understood as following. Since the CHD3 molecule is a symmetric top rotor with the CH bond as the symmetry axis, k0 is a good quantum number. For the initial (1,1) state, the rotational angular momentum has a projection of 1 on the CH axis, or the molecule rotates around the CH bond. In contrast, for the initial (1,0) state, the rotational angular momentum has a projection of 0 on the CH axis, or the molecule rotates around an axis perpendicular to the CH
Jtot=1
2.5
Jtot=21 Jtot=41
(b) 0.0 0.4
0.8
1.2
1.6
Translational energy (eV) FIG. 3. (a) Same as Fig. 2(a) except for (J0 = 1, k0 = 1, K0 = 0) state; (b) Same as Fig. 2(b) except for (J0 = 1, k0 = 1, K0 = 1) state.
bond. When the incoming H atom collides with the CHD3 molecule, because the projection of the orbital momentum on the body-fixed axis is equal to zero, the initial K0 of the system is given by the projection of CHD3 rotational angular momentum on the body-fixed axis. For the initial (J0 ,k0 ,K0 ) = (1,0,0) and (1,1,1) states, the initial projection of total angular momentum on body-fixed axis K0 equal to k0 means the body-fixed axis is parallel to the CH bond, or the incoming H atom approaches along the CH bond. As a result, these initial states have large reaction probabilities because the reaction has a collinear saddle geometry. In contrast, for the initial (J0 ,k0 ,K0 ) = (1,0,1) and (1,1,0) states, the body-fixed axis is perpendicular to the CH bond, or the incoming H atom approaches perpendicular to the CH bond, resulting in much smaller reaction probabilities as compared to the (1,0,0) and (1,1,1) states. In Ref. 36, we calculated the CC integral cross sections for the title reaction by including sufficient numbers of Kblocks for Jtot > 0, and found the CC ICSs are considerably larger than the corresponding CS results, in particular for the ground vibrational state. To investigate the accuracy of the CS approximation for the rotationally excited state, we carried out CC calculations for these rotationally excited initial states. Figure 4 shows the CC total reaction probabilities for the initial state (J0 = 1,k0 = 0). For the K0 = 0 initial state shown in Fig. 4(a), the CC reaction probabilities shift regularly with the increasing of Jtot , slightly slower than the CS results shown in Fig. 2(a), however the CC probabilities for Jtot = 20, 40, in particular for Jtot = 40 are considerably smaller than the CS results in high energy region. Figure 4(c) shows the reaction probabilities for a few Jtot with K0 = 1 and ε = −1 (denoted as K0 = 1− ). As seen, the CC probabilities are slightly larger than the corresponding CS ones, and the thresholds for CC probabilities are also slightly lower. In strong contrast, the CC probabilities for K0 = 1+ shown in Fig. 4(b) behave very differently from the corresponding CS ones in Fig. 2(b), as well
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144309-5
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
Probability(10 )
4.0
5.0
Jtot=0 Jtot=20 Jtot=40
(a)
Jtot=20 Jtot=40
2.0
(a)
Jtot=41
4.0
+
K0=1
-3
Probability(10 )
Jtot=1 Jtot=21
(b)
0.0
Jtot=1 Jtot=21
2.5
Jtot=41
(b)
0.0 Jtot=1 Jtot=21
0.25
Jtot=41
(c)
0 0.4
0.8
-
K0=1
-3
Probability(10 )
-
K0=1
-3
Jtot=0
0.0 +
K0=1
-3
Probability(10 )
0.0
Probability(10 )
K0=0
-3
K0=0
-3
Probability(10 )
10.0
1.2
Jtot=1
2.5
Jtot=21 Jtot=41
(c)
0.0 0.4
1.6
Translational energy (eV)
0.8
1.2
1.6
Translational energy (eV)
FIG. 4. (a) CC total reaction probabilities for the initial state (J0 = 1, k0 = 0, K0 = 0); (b) Same as Fig. 4(a) except for (J0 = 1, k0 = 0, K0 = 1+ ) state; (c) Same as Fig. 4(a) except for (J0 = 1, k0 = 0, K0 = 1− ) state.
FIG. 5. (a) Same as Fig. 4(a) except for (J0 = 1, k0 = 1, K0 = 0) state; (b) Same as Fig. 4(b) except for (J0 = 1, k0 = 1, K0 = 1+ ) state; (c) Same as Fig. 4(c) except for (J0 = 1, k0 = 1, K0 = 1− ) state.
as its K0 = 1− counterparts shown in Fig. 4(c). And the difference increases with the increase of Jtot s. For the first few Jtot , the threshold energies actually decrease with the increase of Jtot . More importantly, the level-off probability values significantly increase with the increase of Jtot . Obviously, the Coriolis coupling between different K blocks in the CC calculations substantially change the reaction probabilities for this initial state, in particular for the K0 = 1+ case. For initial K0 = 0 shown in Fig. 4(a), coupling to the K > 0 blocks reduces the total reaction probabilities, in particular when Jtot is large, because the K = 0 state is much more reactive for the (J0 = 1,k0 = 0) initial state as shown in Fig. 2. On the other hand, for initial K0 = 1+ , the coupling to the K = 0 block substantially enhances reactivity as shown in Fig. 4(b). For initial K0 = 1− , because the rotational basis functions with K = 0 do not exist for k = 0 due to parity constrain as can be seen from Eq. (9), there is no coupling to K = 0 block until k is excited from initial 0 to 3, or larger. As a result, the CC probabilities for K0 = 1− are only slightly larger than the corresponding CS ones. Figure 5 shows the CC total reaction probabilities for the initial state (J0 = 1,k0 = 1). From Fig. 5(a) we can see that the behavior of K0 = 0 initial state is very similar to that for (J0 = 1,k0 = 0, K0 = 1+ ) as shown in Fig. 4(b). Apparently, the coupling to the K = 1 block substantially enhances the reactivity for large Jtot because for this initial state the K = 1 state is much more reactive as shown in Fig. 3. While for the K0 = 1 case, the coupling to the K = 0 block slightly reduces the probabilities for both parities as shown in Figs. 5(b) and 5(c), compared to the corresponding CS results. In order to obtain the integral cross section, we calculated the total reaction probabilities for Jtot up to 65 to converge the ICS for collision energy up to 1.6 eV. In Fig. 6, we present the CS and CC K0 specified cross sections
for the initial state (J0 = 1,k0 = 0). The CS cross sections steadily increase with the increase of collision energy, with σ (K0 = 0) much larger than σ (K0 = 1), reaching 0.27 a02 at Ec = 1.6 eV. The CC σ (K0 = 0) is a little bit larger than the CS σ (K0 = 0) for collision energy up to 0.7 eV. It reaches the maximum value at around 1.1 eV, then begins to decline slowly with the further increase of collision energy. At Ec = 1.6 eV, the CC σ (K0 = 0) is 0.16 a02 , considerably smaller than the CS value. The CC σ (K0 = 1− ) is only slightly larger than the CS σ (K0 = 1), both are negligibly small as compared to their counterparts. In strong contrast, the CC σ (K0 = 1+ ) is much larger than the σ (K0 = 1− ), even larger than the CC σ (K0 = 0) when the collision energy is higher than 1.3 eV. Figure 7 presents the CS and CC K0 specified cross sections for the initial state (J0 = 1, k0 = 1). The CS σ (K0 = 1) is much larger than the σ (K0 = 0), totally opposite to the initial
CS K0=0
2
Cross Section (a0 )
0.25
CS K0=1 CC K0=0 +
0.2
CC K0=1
-
CC K0=1
0.15 0.1 0.05 0 0.4
0.8
1.2
1.6
Translational Energy (eV) FIG. 6. CS and CC K0 specified cross sections for the initial state (J0 = 1, k0 = 0).
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144309-6
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
0.2
0.25
CC K0=1
0.1
CC K0=1
0.2
+ -
0.05
CS K0=2 CC K0=0
2
0.15
CS K0=1
Cross Section(a0 )
2
Cross Section (a0 )
CS K0=0
CS K0=0 CS K0=1 CC K0=0
+
CC K0=1
-
0.15
CC K0=1
+
CC K0=2
-
CC K0=2
0.1
0.05
0 0.4
0.8
1.2
0 0.4
1.6
0.8
1.2
1.6
Translational Energy (eV)
Translational Energy (eV) FIG. 7. Same as Fig. 6 except for (J0 = 1, k0 = 1) state.
FIG. 9. Same as Fig. 6 except for (J0 = 2, k0 = 1) state.
state (J0 =1, k0 = 0) shown in Fig. 6. In contrast, the differences between the CC cross sections for K0 = 0 and 1 are not big, in particular in the high energy region. The CC σ (K0 = 1) for both parities are quite close, in particular in the low energy region. The CC σ (K0 = 0) are considerably smaller than σ (K0 = 1) in the low energy, but surpass σ (K0 = 1− ) at Ec = 1.4 eV. The K0 specified CS cross sections in Figs. 6 and 7 show very strong steric effect of the reaction, i.e., the reactivity strongly dependent on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . The CC σ (K0 = 0) for the (J0 = 1, k0 = 0) initial state is still considerably larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) as can be seen from Fig. 6, in particular in the low collision energy region, therefore rather strong steric effect still persist for that state although as not strong as in the CS case. However, for the (J0 = 1, k0 = 1) initial state the CC σ (K0 = 0) is only slightly larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) as shown in Fig. 7, indicating steric effect is very weak for the state. We also carried out CS and CC calculations for the J0 = 2 initial states with k0 = 0, 1, 2. Due to a large number of reaction probabilities obtained, we only present K0 specified cross sections in Figs. 8–10. From Fig. 8, we can see that for
the (J0 = 2, k0 = 0) initial state the CS K0 specified CS cross sections show very strong steric effect, with σ (K0 = 0) much larger than σ (K0 = 1) and σ (K0 = 2) close to zero, similar to that for the (J0 = 1, k0 = 0) initial state shown in Fig. 6. At the CC level σ (K0 = 0) is larger than the average of σ (K0 = 1+ ) and σ (K0 = 1− ) only in the low collision energy region, while σ (K0 = 2− ) remains negligible compared to its K0 = 0,1 counterparts. For the (J0 = 2, k0 = 1) initial state shown in Fig. 9, the CS σ (K0 = 1) are much larger than the other two. However, the CC cross sections behave very differently. For this initial state, σ (K0 = 1− ) and σ (K0 = 2− ) are larger than their positive parity counterparts. As can be seen from Fig. 10 the CS cross sections for the (J0 = 2, k0 = 2) initial state are very similar to those for the (J0 = 1,k0 = 1) with K0 = k0 state much more reactive. While at the CC level σ (K0 = 2) are larger than the other two only in the low energy region. Therefore, as for the J0 = 1 initial state, very strong steric effects for the J0 = 2 initial state only exist under the CS approximation. Finally, in Fig. 11 we show the CS and CC cross sections for the J0 = 0, 1, 2 initial rotational states. No matter at the CS or CC level, the K0 specified cross sections for the J0 = 1, 2 initial states manifest dramatic differences from the J0 = 0 initial state as shown in Figs. 6–10. However, after K0 0.25
0.4
CS K0=0
CS K0=0
CS K0=1
CS K0=1
Cross Section(a0 )
+
0.3
CC K0=1
-
CC K0=1
+
CC K0=2
-
0.2
CC K0=2
0.1
0 0.4
0.2
CS K0=2 CC K0=0
2
2
Cross Section(a0 )
CS K0=2 CC K0=0
+
CC K0=1
-
0.15
CC K0=1
+
CC K0=2
-
CC K0=2
0.1
0.05
0.8
1.2
Translational Energy (eV) FIG. 8. Same as Fig. 6 except for (J0 = 2, k0 = 0) state.
1.6
0 0.4
0.8
1.2
1.6
Translational Energy (eV) FIG. 10. Same as Fig. 6 except for (J0 = 2, k0 = 2) state.
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144309-7
Z. Zhang and D. H. Zhang
J. Chem. Phys. 141, 144309 (2014)
regardless of its initial k0 , does not have any effect on the total cross sections. It also means that the errors introduced by the CS approximation on ICS for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state.
2
Cross Section(a0 )
CC
0.1 CS
ACKNOWLEDGMENTS
(0,0) (1,0) (1,1) (2,0) (2,1) (2,2)
0.05
0 0.4
0.8
1.2
This work was supported by the National Natural Science Foundation of China (Grant No. 90921014), Ministry of Science and Technology of China (2013CB834601), and Chinese Academy of Sciences. 1.6
Translational Energy (eV) FIG. 11. CS and CC cross sections for the J0 = 0, 1, 2 initial rotational states.
averaging the obtained CS and CC cross sections for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, meaning that the rotational excitation of the initial CHD3 up to J0 = 2 does not have any effect on the total cross sections for the title reaction. It is also interesting to see that the errors introduced by the CS approximation on ICS for the rotationally excited J0 = 1, 2 initial states are the same as those for the J0 = 0 initial state reported in Ref. 36. IV. CONCLUSIONS
The time-dependent wave packet method has been applied to calculate the CS and CC probabilities and cross sections for the title reaction with CHD3 in J0 = 1, 2 rotationally excited initial state under the C3v approximation on a new potential energy surface recently constructed by this group using neural network fitting.48 At the CS level, it is found that the K0 = k0 initial states are much more reactive, meaning that the reactivity strongly dependent on the orientation of the reagent CHD3 with respect to the relative velocity between the reagents H and CHD3 . However, at the CC level this dependence becomes much weak although in general the K0 specified cross sections for the K0 = k0 initial states remains primary to the overall cross sections, indicating the Coriolis coupling between different K blocks are important to the dynamics of the title reaction. This is in agreement with what we found that the CC cross sections are considerably larger than the corresponding CS results due to the strong Coriolis coupling.36 More calculations should be carried out on the other X + YCZ3 type reactions to investigate if the failure of the CS approximation to accurately describe the dynamics is general to the complete X + YCZ3 family. Our calculations revealed that the overall CS and CC cross sections obtained after K0 averaging for the J0 = 1, 2 initial states with all different k0 are essentially identical to the corresponding CS and CC results for the J0 = 0 initial state, despite the fact that the K0 specified cross sections for the J0 = 1, 2 initial states exhibit dramatic differences from the J0 = 0 initial state. This clearly showed that for the title reaction the initial rotational excitation of CHD3 up to J0 = 2,
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