Quantum Dynamics Study of Vibrational Excitation Effects and Energy

Article pubs.acs.org/JPCA

Quantum Dynamics Study of Vibrational Excitation Effects and Energy Requirement on Reactivity for the O + CD4/CHD3 → OD/OH + CD3 Reactions Wei Yan, Fanbin Meng, and Dunyou Wang* College of Physics and Electronics, Shandong Normal University, Jinan 250014, China ABSTRACT: A quantum reactive dynamics, six-degrees-of-freedom, time-dependent wave packet method is employed to study vibrational enhancement and energy requirement on reactivity of the O(3P) + CD4/CHD3 → OD/OH + CD3 reactions. The calculations show, for O + CD4, that all the vibrational excitations of CD4 enhance reactivity, which agrees with quasi-classical trajectory results. However, this finding contradicts the experimental observation where the bending excitation suppresses reactivity. The present study also reveals that translational energy, in general, is more effective to enhance reactivity than vibrational energy; however, at higher collision energy, vibrational energy is slightly more effective than translational energy. For O + CHD3, the stretching and bending excitations of CHD3 enhance the reaction, whereas the umbrella motion hinders reactivity. The calculated excitation functions agree well with experiments.

1. INTRODUCTION The reactions of Cl, F, and O atoms with methane recently have been the focus of experimental1−10 and theoretical investigations11−29 to study mode-selective enhancement and energy requirement on reactivity. The Polanyi rules,30 which were summarized on the basis of the atom−diatom reaction systems, state that for an endoergic late barrier reaction on the potential energy surface (PES), the reactant vibrational energy is more effective than translational energy in surmounting the energy barrier into products; the reverse is true for an exoergic early barrier reaction. However, the extension of the Polanyi rules to more complex polyatomic reaction systems is not clear. The above reactions have become the stereotype reactions to investigate energy requirements on reaction reactivity on more than four atom polyatomic reaction systems. The original experimental work of Cl + CHD3 by Yan et al. in 20071 found that C−H vibrational motion is no more effective than the translational energy for the Cl + CHD3 reaction even though it has a very late barrier, which contradicts the Polanyi rules for a later barrier reaction. However, later in 2012, two quantum dynamics studies, one a six-degrees-offreedom (DOF) study on the Cl + CH411 and another a 7 DOF study on the Cl + CHD312 reaction, showed that only at very low collision energies do the Polanyi rules break. In 2013, Wang et al. redid the experiment2 by probing all rotational channels of CD3 (ν = 0), and the new results agree with theoretical observations. For the F + CH4 reaction with an early barrier on the PES, experimental studies6 on the F + CHD3 find that one quanta excitation of the CH stretch hinders the reaction rate and that the DF + CHD2 channel is the favored product channel. This feature has been studied and explained by a quasiclassical trajectory (QCT) calculation.16,17 Also, resonances have been © 2013 American Chemical Society

found near the reaction threshold by experiments for both the F + CH4 and the F + CHD3 reactions.4,5 This finding has been confirmed by a recent reduced-dimensionality quantum dynamics study.18 The O + CH4 reaction, with a more-or-less center barrier, plays an important role in combustion processes.31 Zhang and Liu measured the differential cross sections for the O(3P) + methane reaction in 2005.8 Crossed molecular beam experiments were reported for the reactions of O(3P) with CH4,9 CHD3,10 and CD48. The experimental data show that the excitation function of the O + CH4 increases smoothly with the collision energy;9 the excitation of the C−H vibration in CHD3 enhances reactivity for the OH + CD3 channel;10 however, bending excitations in CD4 slightly restrain the reaction.8 There have been many theoretical studies on this reaction system including the QCT level19−22 and reduced-dimensional quantum dynamics calculations,23−29 but the PESs that have been used are either semiempirical32 or reduced-dimensional ab initio33 PESs. Recently, a full-dimensional ab initio PES has been developed by Czakó and Bowman,34 and their QCT calculations on the O + CHD334 show that C−H stretching excitation can enlarge the reactive cone of acceptance, which increases the OH + CD3 products. In addition, an eightdimensional (8D) quantum dynamics calculation also on this PES was reported35 along with full-dimensional QCT calculations. The comparison of their computed integral cross section (ICS) of the ground-state agrees well with experimental results. They also found that translational energy was more efficient in promoting the reaction than vibrational energy, and Received: September 10, 2013 Revised: October 23, 2013 Published: October 23, 2013 12236

dx.doi.org/10.1021/jp4090298 | J. Phys. Chem. A 2013, 117, 12236−12242

The Journal of Physical Chemistry A

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total angular momentum operator, jρ⃗ is the rotational angular momentum operator for pseudodiatom CX, l ⃗ is the orbital angular momentum operator of the H atom with respect to CX, j ⃗ is the total angular momentum of HCX coupled by jρ⃗ and l,⃗ μr is the reduced mass between CX and H/D, and r is the distance from the center of mass of CX to H/D atom. The moment of inertia, I(ρ), is given by

the stretching excitations had a greater impact than bending and umbrella excitations. More recently, QCT calculations for O(3P) + CH4/CD4 were reported by Czakó et al.36 The results show that the reactant bending excitations slightly enhance the reactivity, whereas stretching excitations activate the reaction more efficiently. Because there have been no quantum dynamics studies reported so far on the O + CD4/CHD3 reactions, in this paper, we performed the first quantum dynamics study with a sixdegrees-of-freedom (6DOF), time-dependent wave packet quantum scattering approach for the title reaction on the full dimensional PES by Czakó and Bowman.34 We treated the three nonreactive D atoms as a pseudo atom, and so the system becomes an atom−triatom reaction system. Our goals were to study the vibrational specified state enhancement on reactivity, to compare our computed integral cross sections with QCT and experimental measurements, and to investigate energy efficiency on reactivity for the title reaction systems.

⎛ ρ⎞ I(ρ) = 3mDρ2 sin 2 η ; η = cos−1⎜⎜ ⎟⎟ ⎝ ρe ⎠

Here, mD is the mass of D atom, ρ is the vibrational distance from C to X atom, and ρe is the distance of the CD equilibrium geometry. The vibrational reference Hamiltonians hr(r) and hρ(ρ) are given as

2. THEORETICAL METHODS We employed a six-degrees-of-freedom reduced dimensional quantum dynamics study on the O + CD4/CHD3 reaction. In our 6DOF model,11,37 the three nonreactive D atoms in CD4/ CHD3 were treated as one pseudoatom, X, located at the three D atoms’ center of mass. The pseudoatom X moving toward or away from the C atom corresponds to the three D atoms moving in phase to open or close the umbrella cone, while the three C−D distances are fixed at the asymptotic bond length. Thus, this motion approximately represents the umbrella motion of CD3.Yu and Nyman originated this C−X umbrella approach in their 3DOF Cl + CH4 quantum scattering calculation38 and used it again in later studies on H + CH4,39 Cl + CH4,40 and F + CH4.41 The 6DOF Hamiltonian in the reactant Jacobi coordinates, shown in Figure 1, can be written as

hr (r ) = −

ℏ2 ∂ 2 + V1D(r ) 2μr ∂r 2

hρ(ρ) = −

ℏ2 ∂ 2 + V1D(ρ) 2μρ ∂ρ2

(3)

Here, V1D(r) and V1D(ρ) are 1D reference potentials for r and ρ. These potentials are cuts in the respective coordinates with other coordinates fixed at the equilibrium geometry. The split-operator method42 is employed here to propagate the wave packet on the PES for the quantum dynamics process. The time-dependent wave function can be expanded in terms of the body-fixed (BF) rovibrational eigenfunctions defined in terms of the reactant Jacobi coordinates. After the wave function is propagated into the product region, the standard time-dependent reactive flux43−46 is employed to extract the initial-state-selected reaction probability. To obtain the integral cross section, we calculate the partial wave reaction probabilities of the total angular momentum J. The initial-state-selected integral cross section συ0,j0(E) is obtained by summing over all the initial-state-selected reaction probability PJυ0,j0,K0 for all different partial waves

2

j ρ⃗ (J ⃗ − j ⃗ ) ℏ2 ∂ 2 + + hr (r ) + hρ(ρ) + H=− 2 2 2μ ∂z 2I(ρ) 2μz 2

l⃗ + + V6D(z , r , ρ , θ , γ , φ) 2μr r 2

(2)

συ0j (E) =

(1)

0

Here, μ is the reduced mass of the whole system, z is the distance from the center of mass of HCX to the O atom, J ⃗ is the

1 π 2j0 + 1 k 2

∑ (2J + 1)PυJ j K (E) 00

J

0

(4)

where E is the translational energy, υ0 denotes the initial vibrational quantum number, j0 denotes the initial rotational quantum number, K0 is the projection of J onto the BF z axis of the atom−triatom system, and k is the wavenumber. For the translational coordinate z in the range from 3.2 to 12.2 bohr, we use 145 sine basis functions to expand the wave function, among these 50 are used to expand the wave function in the interaction region. Twenty-six potential-optimized vibrational discrete variable representation(DVR) points47 are used for r coordinates in the range from 1.3 to 6.0 bohr; 7 potential-optimized vibrational DVR points are used for CX covering the range from 0.0 to 1.5 bohr. Twenty-four spherical harmonic rotational functions are used for θ and 31 for γ, which are coupled to give 7800 parity adapted total angular momentum basis. The time-dependent wave packet is propagated with a time step of 15 a.u. for a total time of about 7500 atomic unit time. The above numerical parameters converge the calculations for the current 6DOF reduceddimensional dynamics calculation for both the O + CD4 and the O + CHD3 systems.

Figure 1. Reactant Jacobi coordinates for the reactions O + CD4/ CHD3 → OD(H) + CD3. 12237



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3. RESULTS AND DISCUSSION (1). O + CD4 → OD + CD3. A. Integral Cross Sections. Figure 2 shows the reaction probability for six different partial

barrier height; thus, it decreases the reaction threshold and results in unreal tunneling effects. B. Comparison of the ICSs. Figure 4 compares the ICSs for the excited vibrational states between the present calculation

Figure 2. Reaction probabilities of ground-state CD4 for different partial waves J = 0, 20, 40, 60, 80, and 100 as a function of collision energy.

waves, J = 0, 20, 40, 60, 80, and 100 for the ground vibrational state (0, 0, 0). The integral cross section was computed using the J-shifting approximation48 with a J interval of 10. In total, 140 partial wave reaction probabilities were needed to converge the ICS for the ground vibration state (0, 0, 0) which is shown in Figure 3. The comparison of the ICS between the 6DOF

Figure 4. Comparison of the integral cross sections for the O + CD4 reaction with the QCT results36 [ν4, ν2, and ν1 are the umbrella, bending, and symmetric stretching modes of CD4 in QCT]. (0, 0, 1), (0, 1, 0), and (1, 0, 0) label the umbrella, bending, and symmetric stretching modes of CD4, respectively.

and the QCT study.36 To converge the ICSs for the umbrella (0, 0, 1) (as ν4 = 1 for QCT), bending (0, 1, 0) (as ν2 = 1 for QCT), and stretching mode (1, 0, 0) (as ν1 = 1 for QCT) of CD4, 140, 150, and 155 partial waves are computed to obtain the 6DOF quantum ICSs, respectively. The 6DOF quantum results and the QCT results36 agree with each other very well for the ground state (ν = 0 for QCT) and umbrella motion (ν4 = 1 for QCT). The ICSs for the stretching motion and bending motion are larger than for the QCT results. This might be because our model is a 6DOF one while the QCT is full dimensional. In addition, the different descriptions of excited vibrational motions were used in these two calculations. In our model, we used the Jacobi coordinate, and so the excited vibrational motions are not exactly the same as that of a normal mode motion in the QCT description. Nonetheless, both the 6DOF quantum and the QCT results show that all the excited vibrational motions enhance this reaction’s reactivity in the following order: stretching > bending > umbrella > ground state. Among the three excited states, the stretching motion and the bending motion enhance the reactivity the most. These findings agree with the previous reduced-dimensionality quantum-scattering results23−28,50,51 which show that either the stretching or the bending motion of CD4 enhances reactivity. However, this finding is in sharp contrast to the experimental observation of the O + CD4 by Zhang and Liu8 who found that the bending exited motions (either the bending or the umbrella motion as they cannot distinguish between the two) of the CD4 hinder the reactivity. The experiment does not probe all the CD3 product states; thus, there is uncertainty existing in the experimental data. Nonetheless, because we only did a 6DOF reduced dimensional quantum dynamics calculation and because the QCT calculation does not properly account for the quantum effects, this might produce the discrepancy between the theoretical and experimental data. In future studies, a more reliable comparison with the experimental data would be based on more dimensional or full dimensional quantum dynamics calculation or on QCT

Figure 3. Comparison of the ICS for the O + CD4 reaction between the 6DOF quantum dynamics and the QCT result36 of the ground vibrational state.

quantum dynamics and the QCT results36 for the ground vibrational state is plotted in Figure 3 as a function of collision energy. The two results agree with each other very well. The 6DOF quantum result shows a reaction threshold of about 10.4 kcal/mol, which is lower than the ground-state adiabatic height 11.9 kcal/mol. This indicates considerable tunneling effect. However, this tunneling effect is smaller than that in the O + CH4 case,35 where the 8DOF quantum result shows a threshold at about 8.0 kcal/mol while the adiabatic barrier height is at 11.0 kcal/mol. This is not surprising since D atom is much heavier than H atom, which leads to the O + CD4 reaction producing a smaller tunneling effect. As shown in Figure 3, the QCT result also shows a tunneling effect and a lower threshold in the comparison. This unreal tunneling effect is from the zero point energy (ZPE) leak feature in the QCT calculation,49 which causes a lower vibrationally adiabatic barrier height. The QCT calculation uses all the trajectories without the ZPE constraints. As a result, QCT does not follow a vibrationally adiabatic PES, which can lead to a lower vibrationally adiabatic 12238



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studies that include proper zero-point energy and tunneling effects. C. Energy Efficiency on Reactivity. To investigate the energy efficiency on reactivity for the O + CD4 reaction, the ICS ratio of the C−D stretching motion σ (1, 0, 0) and ground-state σ (0, 0, 0) is plotted in Figure 5 on the basis of an equivalent

Figure 6. Contour plot of the PES in terms of Z (the distance from O to the center of mass of CD4) and r (the distance from D to the center of mass of C−X). All other degrees of freedom are fixed at transitionstate geometry.

Figure 5. Cross section ratio of the ICS for the O + CD4 reaction, σ (1, 0, 0)/σ (0, 0, 0), between the C−D stretching mode and the ground state as a function of collision energy on the basis of an equal amount of total energy, where the vibrational energy difference Ev between the two states is 6.3 kcal/mol.

amount of the total scattering energy. As seen, for collision energy less than 11.0 kcal/mol, the ratio has a rapid, steep rise before it reaches 1, and translational energy is the dominant force in raising the reactivity. Ratios greater than 1 at high collision energy are only slightly larger than 1, which means vibrational energy here is only slightly more efficient than translational energy on promoting the reaction. Overall, for this O + CD4 reaction, translational energy is more efficient than vibrational energy on reactivity, which agrees with the QCT observation.36 Usually for this reaction, the transition state is viewed as a more-or-less center barrier;10,35 however, because the reaction for the current PES is 5.8 kcal/mol endoergic, the transition state actually has a slightly late barrier. It can be verified from the contour plot of the PES, in terms of Z (the translational coordinate from the O atom to the center of mass of CD4) and r (the stretching coordinate from D to the center of mass of CX), that the transition state, at Z = 4.67 bohr, r = 2.56 bohr, already slightly lies in the product channel in Figure 6. Thus, from a slightly late barrier point of view, an intuitive application of the Polanyi rules would predict that vibrational energy should be more efficient on reactivity than translational energy. This is obviously not the case here. However, the Polanyi rules are derived from the atom−diatom reaction systems with apparent early and late transition states, which should not be directly extended for the present polyatomic O + CD4 reaction because it presents a more-or-less center barrier. These results further prove the complexity of the energy requirement on reactivity for polyatomic reactions. Combined theoretical and experimental investigations are needed to better understand the distinctive phenomena presented in polyatomic reactions. (2). O + CHD3 → OH + CD3. A. Comparison of the ICSs. Figure 7 compares the ICSs for four different initial vibrational states of CHD3 with the available QCT results for the ground state (ν = 0) and the symmetric stretch (ν1 = 1).34 In total, 155, 160, 165, and 175 partial waves are needed to converge the initial-state-selected ICSs for the ground (0, 0, 0), umbrella (0, 0, 1), bending (0, 1, 0), and symmetric stretching (1, 0, 0)

Figure 7. Comparison of the integral cross sections for the O + CHD3 reaction with the quasiclassical trajectory results34 [ν1 is the symmetric stretching mode of CHD3]. (0, 0, 1), (0, 1, 0), and (1, 0, 0) are the umbrella, bending, and symmetric stretching modes of CHD3, respectively.

modes of CHD3, respectively. The 6DOF ICS agrees well with the QCT results for the ground state but not very well for the excited stretching motion. This discrepancy on the excited stretching motion is due to different dimensionality and description on the vibrational coordinates, which was explained in the O + CD4 case above. The stretching motion of C−H (1, 0, 0) and the bending motion (0, 1, 0), again similar to the O + CD4 reaction, enhance the reactivity. However, the biggest difference here, contrary to the O + CD4 case, is that the umbrella excitation of CHD3 hinders the reactivity as the (0, 0, 1) ICS is smaller than that of the ground state. This phenomenon is understandable because the umbrella motion here favors the OD + CHD2 product channel instead of the OH + CD3 one as the umbrella excitation drains the energy from the C−H stretching motion and prohibits the breaking of the C−H bond, thus hindering the forming of OH + CD3 product channel we study here. B. Energy Efficiency on Reactivity. Figure 8 gives the ICS ratio, σ (1, 0, 0)/σ (0, 0, 0), of the stretching motion to the ground state on the basis of the equal amounts of total energy. This figure shows, for collision energy lower than 7.1 kcal/mol, that translational energy is more effective to promote the reaction than vibrational energy. At high energy larger than 7.1 12239



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Second, our calculation does not distinguish the product states; however, the experimental result is only for the ground product state, and so there is a possibility that the good agreement between the experimental results and our 6DOF results is due to an error cancellation.

4. CONCLUSIONS We have carried out reduced-dimensional quantum reaction dynamics, time-dependent wave packet propagation computations to study the O + CD4/CHD3 reaction systems. A sixdegrees-of-freedom approach is employed to study the vibrational mode selected enhancement and energy requirement on reactivity on the ab initio PES developed by Czakó and Bowman.34 For the O + CD4 reaction system, we found that all the vibrational excitations, including bending motion, enhance the reactivity, which agrees with the QCT calculation.36 This finding contradicts the experimental result by Zhang and Liu8 who state that the bending excitation of CD4 suppresses the reactivity. As noted, their measurements only probe the CD3(ν = 0) product. In addition, translation energy increases reactivity more effectively than vibrational energy in the overall reaction. For the O + CHD3 system, the 6DOF results show that the stretching motion and the bending motion of CHD3 enhance the reactivity while the umbrella motion hinders the reactivity. When the collision energy is lower than 7.1 kcal/mol, translational energy is more effective promoting the reaction than vibrational energy; above that, the reverse is true. The comparison of the ICSs between our results and the experimental excitation function10 shows that the ICSs agree with each other quite well. Overall, the Polanyi rules which were derived from the atom−diatom reaction cannot be simply extended to the more complex reaction system, O + CD4/CHD3. For the O + CD4 reaction with a slightly late barrier on the PES, translational energy actually is more efficient on reactivity than vibrational energy. Our results on bending excitation agree with the QCT result on promoting the reactivity for the O + CD4 reaction, which contradicts the experimental observation. The difference between the theoretical calculations and the experimental results needs further investigations to resolve these inconsistencies. These results further prove both theoretical and experimental investigations are needed to better understand the more complex polyatomic reaction systems.

Figure 8. Cross section ratio of the ICS for the O + CHD3 reaction, σ (1, 0, 0)/σ (0, 0, 0), between the C−H stretching mode and the ground state as a function of collision energy on the basis of an equal amount of total energy, where the vibrational energy difference Ev between the two states is 8.4 kcal/mol.

kcal/mol, vibrational energy is more effective than translational but not by much because the biggest ratio is only around ∼1.6. This ratio is larger than that of ∼1.2 for O + CD4 at higher energy part, which means that C−H stretching is more efficient than C−D stretching in promoting the reaction rate at the high energy. This is true because H atom is lighter than D atom; thus, CHD3 is more active than CD4 and has a larger reaction rate with an isotope effect around 9.1. C. ICS Comparison with Experiment. In Figure 9, we compare our ground-state and excited stretching mode ICSs

Figure 9. Comparison of the integral cross sections for the O + CHD3 reaction with the experimental results10 [ν1 is the symmetric stretching mode of CHD3]. Our data is multiplied by 2.4 to help compare.

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with the measurements of the excitation functions by Wang and Liu.10 Because the experiment does not report absolute cross sections, we rescaled our computed ICS at the highest experiment point of 13.0 kcal/mol to compare the data with the measured excitation functions. Because only the groundstate CD3 products were detected for the experimental measurement and because the signals were too weak to be quantified for the vibrationally excited CD3 products, the CD3 (ν = 0) channel is the major product. Nonetheless, this figure shows that theoretical predictions and experimental measurements are in excellent agreement at the threshold energy, ∼8.0 kcal/mol for ground state and ∼4.5 kcal/mol for the C−H stretching mode, respectively; also, the shapes of the 6DOF agree well with the experimental measurements. A couple of issues need to be addressed regarding this comparison. First, the reduced-dimensionality nature of this calculation might result in an uncertainty error in the cross section calculation.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.Y. Wang thanks Dr. G. Czakó for sending us the PES and insight information on the topic. D.Y. Wang also thanks Drs. F. Wang and K. Liu for providing us their experimental data. This work is supported by the National Natural Science Foundation of China (Grant No. 11074150) and Taishan Scholarship fund. The computation work was carried out at the Shenzhen Supercomputer Center in China. 12240



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