Coriolis Coupling Effects in O+(4S) + H2(X1Σg+ ... - ACS Publications

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Coriolis Coupling Effects in O+(4S) + H2(X1Σ+g) → OH+(X3Σ−) + H(2S) Reaction and Its Isotopic Variants: Exact Time-Dependent Quantum Scattering Study Wenwu Xu, Wenliang Li, Shuangjiang Lv, Hongsheng Zhai, Zhixin Duan, and Peiyu Zhang* State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, China S Supporting Information *

ABSTRACT: The time-dependent wave packet quantum method taking into account the Coriolis coupling (CC) has been employed to investigate the dynamics of O+ + H2/D2/ HD (vi = 0, ji = 0) reactions based on an accurate potential energy surface [Martínez et al. J. Chem. Phys. 2004, 120, 4705]. Through the comparison between the results with and without CC, the pronounced CC effects have been revealed in the title reactions. Moreover, the calculated results with the CC method can well reproduce the data of close-coupling hyperspherical (CCH) exact quantum method. The calculations demonstrate that the CC effects play an important role in the O+ + H2 system. et al.26 to investigate the dynamics of the title reactions. The results demonstrated the agreement between the theoretical calculations and experimental measurements. However, under the helicity decoupling or centrifugal sudden (CS) approximation, which neglects the coriolis coupling (CC) terms of the Hamiltonian operator, the HD-RWP reaction probabilities and cross sections cannot well reproduce the CCH results. Actually, the CC effect, which is an important issue in chemical reaction dynamics,1,11,27−40 has been an attractive subject to theoretical investigations in recent years. Chu et al. have observed pronounced CC effects through the theoretical investigations on a series of ion−molecule reactions. In this context, it is desirable to accurately study the dynamics of O+ + H2 reaction and its isotopic variants taking into account the CC effect. Here, both the CC and CS calculations using the timedependent wave packet method are carried out on the O+ + H2 reaction and its isotopic variants by employing an accurate ́ et al.23 It is shown that the potential energy surface of Martinez time-dependent CC results for the O+ + H2 reaction are in agreement with those reported in ref 25 at the timeindependent CCH level, because both methods correspond to exact treatments of the dynamics and the same PES was used. In addition, the time-dependent CS results for O+ + H2 reaction and its isotopic variants D2 and HD are accordant with those reported in ref 26 at the RWP-HD level, because the same quantum approximation (CS or HD) was used in both cases. The only difference between time-dependent CS method

I. INTRODUCTION Ion−molecule reactions have attracted considerable attention in recent years because of their important role in the collision processes of interstellar media, plasmas, planetary ionospheres, and high-energy physics studies.1−21 High-quality experiments and a variety of dynamics simulations of the O+ + H2 reaction have been performed to deeply understand the ion−molecule reaction dynamics. On the experimental side, Burley et al.22 measured the kinetic energy-dependent rate constant, cross sections, and branching ratios of ground-state atomic oxygen ion with molecular hydrogen and its isotopic variants D2 and HD using guided ion beam mass spectrometry at a reagent rovibrational temperature of 300 K. ́ Theoretically, Martinez et al.23 reported the best ab initio analytical potential energy surface (PES) available using CCSD(T) method with the cc-pVQZ basis set. From the PES, we can determine that the OH+ + H products are accessible from reactants along a collinear minimum energy reaction path, and the PES has a shallow minimum in the products valley. Based on this accurate PES, the quasi-classical trajectory (QCT)24 method was employed to calculate the cross sections of O+ + H2 reaction and its isotopic variants at a very broad collision energy range (ET = 0.01−6 eV), yielding ́ et data in good agreement with experimental results. Martinez al.25 subsequently investigated the title barrierless reaction using the close-coupling hyperspherical (CCH) exact quantum method, and the results were compared with QCT calculations to determine the importance of quantum effects. They found in general a rather good agreement between both methods for the OH+ vibrational and rotational distributions. In 2006, the timedependent real wave packet method using the helicity ́ decoupling (HD-RWP) approximation was used by Martinez © 2012 American Chemical Society

Received: June 7, 2012 Revised: September 13, 2012 Published: October 17, 2012 10882

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III. RESULTS AND DISCUSSION The details of parameters employed in the quantum calculations of O+ + H2 reaction are listed in Table 1. Because

and RWP-HD method is that conventional wave packets were employed here whereas real wave packets were used in ref 26. The paper is organized as follows: In section II the theoretical method and calculation details are described. The results and discussion are considered in section III. Finally, the conclusions are given in section IV.

Table 1. Parameters for the Quantum Calculations of Reaction O+ + H2 (All Quantities in au, unless Otherwise Indicated)

II. CORIOLIS-COUPLED TIME-DEPENDENT QUANTUM DYNAMICS METHOD The Schrödinger equation is written as iℏ

∂ Ψ = Ĥ Ψ ∂t

center of initial wave packet on scattering coordinate width parameter of wave packet average translation energy/eV scattering coordinate (R) range number of translational basis functions number of vibrational basis functions number of rotational basis functions internal coordinate (r) range propagation time

(1)

Using the reactant Jacobi coordinates, the Hamiltonian operator for O+ + H2 is given by 2 ĵ ℏ2 ∂ 2 ℏ2 ∂ 2 ( , , ) − + + θ Ĥ = − V R r 2μR ∂R2 2μr ∂r 2 2μr r 2

+

the numerical values of the parameters we choose for O+ + H2 reaction are relatively large, we use the same set of parameters for the O+ + D2 and O+ + HD reactions except for the width and average translation energy of initial wave packet. The initial vibrational and rotational quantum numbers of H2, D2, and HD in all of our calculations are chosen as vi = 0 and ji = 0, even though this does not exactly correspond to the conditions of the experiments (rovibrational temperature of 300 K). We first investigate the NK-dependent convergence for the reaction probability of O+ + H2 reaction at total angular momentum J = 40. NK denotes the number of K states used in the CC calculation. We have employed a range of different NK values in the calculations. As seen in Figure 1, the results for NK

(J ̂ − j ̂)2 2μR R2

26.0 0.73 0.152 0.1−30 320 150 90 0.5−18 120 000

(2)

where R is the distance from the O+ atom to the center-of-mass of H2, r is the H2 bond length, θ is the angle between R and r, μR is the reduced mass of O+ with respect to H2, μr is the reduced mass of H2, J ̂ is the total angular momentum operator, j ̂ is the rotational angular momentum operator of H2, and V(R,r,θ) is the interaction potential (i.e., the potential energy surface). The initial wave packet is expanded in terms of the bodyfixed (BF) translational−vibrational−rotational basis uvn(R) ̂ ϕv(r) YJMε jK (R,r̂), where n and v are the indices labeling the translational and vibrational eigenfunctions, respectively, and M and K are the projection quantum numbers of the total angular momentum J ̂ on the space-fixed z-axis and BF z-axis, respectively, and ε is the parity of the system. On this basis, the element of the centrifugal term or the Coriolis coupling matrix is expressed as ℏ2 JMε ε ⟨YjK |(J ̂ − j ̂)2 |YjJM ′K′ ⟩ 2 2μR R =

ℏ2 δ {[J(J + 1) + j(j + 1) − 2K 2]δKK ′ 2 jj ′ 2μR R + + − − − λJK λjK (1 + δK 0)1/2 δK + 1, K ′ − λJK λjK (1 + δK1)1/2

δK − 1, K ′} ± λAB

(3)

where = [A(A + 1) − B(B ± 1)] . In the centrifugal sudden (CS) approximation, the off-diagonal elements are neglected, leading to a simplified expression of eq 3, 1/2

Figure 1. Reaction probability dependence with collision energy ET of reaction O+ + H2 at total angular momentum J = 40: (−·−) NK = 1 and 4; (---) NK = 2 and 5; () NK = 3 and 6; (○) CCH (ref 25).

2

ℏ ε JMε ⟨YjK |(J ̂ − j ̂)2 |YjJM ′K′ ⟩ 2μR R2 =

ℏ2 δjj ′[J(J + 1) + j(j + 1) − 2K 2]δKK ′ 2μR R2

= 1 significantly underestimate the exact time-independent results of CCH calculations (hollow circle). With the increase of the maximum NK value investigated, the results gradually approach to the exact CCH results. The marked difference between the CC and CS calculations demonstrates that the CS approximation is not sufficient to describe the dynamics accurately. We further test the NK-dependent convergence at

(4)

The detailed time-dependent wave packet theory used in the calculations was well explained in relevant reviews by Chu et al.1,11 and the references therein. Here we will not go into more details about this. 10883

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Figure 2. Reaction probability dependence with collision energy ET of reaction O+ + H2 at total angular momentum J = 10, 20, 30, and 40: () CC; (−·−) CS; (○) CCH (ref 25); (···) HD-RWP (ref 26).

Figure 3. Reaction probability dependence with total angular momentum J of reaction O+ + H2 at collision energy ET = 0.300, 0.438, and 0.638 eV: () CC; (−·−) CS; (○) CCH (ref 25); (●) HD-RWP (ref 26).

Figure 4. Reaction probability dependence with collision energy ET of reaction O+ + H2 at total angular momentum J = 50, 55, 60, 65, and 70: () CC; (−·−) CS.

J = 50, 60, and 70 in Figure S1 (see the Supporting Information). We find that at larger total angular momentum there is little difference in probabilities between NK = 6 and NK

= 7. Therefore, the convergence of the calculated reaction probability has been achieved for NK = 6 for the O+ + H2 reaction, as it was also obtained in the CCH calculations.25 The 10884

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cross section for reaction O+ + H2 at four collision energies of 0.1, 0.4, 0.7, and 1.0 eV. The involvement of a large number of partial waves is a clear indication of a complex-forming mechanism for the reaction. The oscillations in the contributions gradually become weaker with the increase of collision energy. As is clear by comparing the CC and CS results, the omission of Coriolis coupling has negligible effect on this contribution for low J but has a significant effect with increasing J, as expected. It is also clear that the CS calculations usually have larger contribution and stronger oscillations than the CC ones for high J. This is further supported by the comparison of the integral cross sections of reaction O+ + H2 in Figure 7. The wave packet propagation is performed for J = 0− 80 in calculations. As shown in Figure 7a, both of the CS and HD-RWP cross sections cannot reproduce the CCH ones. The CCH and CC cross sections were slightly larger than the CS and HD-RWP ones at ET < 0.4 eV, whereas the CS and HD-RWP methods give higher cross section values at ET > 0.4 eV. Besides, the experimental results22 of reaction O+ + H2 (300 K) are also represented in the figure as the highest and lower data, which correspond to a 20% experimental error, as it happens for the CCH and HD-RWP results. Both our CC and CS cross sections are within experimental error margins. When the collision energy is lower than 0.5 eV, we can find oscillatory character in CC, CS, and HD-RWP cross sections. Moreover, the CS and HD-RWP cross sections present stronger oscillatory comparing to CC ones, which is in accord with our previous conclusions from the investigations of reaction probabilities. In Figure 7b, we plot the integral cross sections of reaction O+ + H2 in logarithmic scales. The experimental values22 show a significant slope change at ET around 0.3 eV. From Figure 7b, we can observe that the CC cross sections can well reproduce the slope change that occurs at 0.3 eV. Martinez et al. have presented detailed investigations24 using the QCT method on the slope change. They examined the trajectories and concluded the slope change in the excitation function arises from the influence of the absolute PES minimum. The integral reaction cross sections of O+ + D2 reaction are shown in Figure 8. In CC and CS calculations the J values required were J = 0−100. The maximum J to be considered in the O+ + D2 reaction is significantly larger than that in O+ + H2 reaction because of the change of reactant mass. As shown in Figure 8, results similar to those for the integral cross section of O+ + H2 reaction can be found. The CS approximation slightly underestimates the cross sections at ET < 0.35 eV, whereas leads to the overestimation of the cross sections at ET > 0.35 eV. The oscillatory character can be found when the collision energy is lower than 0.4 eV. With the increase of the collision energy, the oscillation tends to disappear. The CC and CS reaction probabilities dependence with collision energy at different J of both O+ + HD → OH+ + D and O+ + HD → OD+ + H reactions are presented in Figures S12 and S13 (Supporting Information), respectively. Results similar to those for the O+ + H2 reaction can be found at both channels. The oscillation in the CS calculations is stronger than that in the CC calculations. And the difference between CC and CS calculations becomes increasingly obvious with the augment of J. For the OH+ + D channel, the probabilities at a majority of J increase sharply to a maximum value, and then decrease slowly, whereas for the OD+ + H channel, the probabilities gradually increase. This phenomenon can be further proved by

results in Figures S2 and S3 (Supporting Information) also indicate that the converged cross sections can be obtained using NK = 6 for O+ + D2/HD reactions. To save computational resources but without sacrificing the computational accuracy, we choose this maximum value of NK = 6 as the optimal number for performing accurate quantum scattering calculations on O+ + H2 reaction and its isotopic variants D2 and HD. The calculated CC and CS reaction probabilities for reaction O+ + H2 at total angular momentum J = 10, 20, 30, and 40 are shown in Figure 2 and are compared with the CCH25 and HD́ RWP26 probabilities of Martinez et al. It is observed that the CC and CS results are in good agreement with CCH and HDRWP ones, respectively, as expected. Similar results can also be found in the O+ + D2 and O+ + HD reactions. The oscillatory character of the reaction probability likely arises from the potential well in the O+ + H2 system. Comparing the CC probabilities with CS and HD-RWP ones, we find that the oscillation in the CS and HD-RWP calculations is stronger than that in the CC and CCH calculations, as is generally expected. This discrepancy in the behavior of oscillation can be attributed to the effect of CC that is the further washing out effects on the resonance probabilities by including different K states in the CC calculation. A similar situation can be found in Figure 3, where J-dependent reaction probabilities are given at three different collision energies of 0.300, 0.438, and 0.638 eV. To further investigate the influence of CC on reaction probability, we present the reaction probabilities as a function of collision energy at J = 50, 55, 60, 65, 70, and 75 in Figure 4. From Figures 2 and 4 we find that the oscillatory features are quite sharp for J < 20, suggestive of long lifetimes, and the resonance widths typically increase with the collision energy and with J. The shifting of the reaction threshold for large J is entirely due to the centrifugal barrier. The difference between CC and CS calculations becomes increasingly obvious with the augment of the total angular momentum J. This impression is further reinforced for the distribution of reaction probability in Figure 5. The horizontal and vertical axes denote collision

Figure 5. Distribution of reaction probability of O+ + H2 reaction: (a) CS; (b) CC. The horizontal and vertical axes denote collision energy ET and J, respectively.

energy and J, respectively. As seen clearly from the figure, the oscillations of the reaction probabilities of almost all J in CS is stronger than that in CC, and the reaction probabilities in CS are lager at higher collision energy (ET > 0.4 eV) and at larger J (J > 50), which are consistent with the results in Figures 2 and 4. In Figure 6, we plot the J-dependent partial wave contributions (weighted over a 2J + 1 factor) to the integral 10885

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Figure 6. Weighed partial wave contributions to the integral cross sections of O+ + H2 reaction as a function of total angular momentum J at four collision energies 0.1, 0.4, 0.7, and 1.0 eV: () CC; (−·−) CS.

reactions are investigated in Figure 9. The wave packet propagation is performed for J = 0−100 in calculations. From

Figure 7. Integral reaction cross section of reaction O+ + H2: (a) linear scales; (b) logarithmic scales; () CC; (−·−) CS; (○) CCH (ref 25); (···) HD-RWP (ref 26); (---) expt (ref 22).

Figure 9. Integral reaction cross section: (a) O+ + HD → OH+(OD+) + D(H); (b) O+ + HD → OH+ + D; (c) O+ + HD → OD+ + H; () CC, (−·−) CS; (···) HD-RWP (ref 26); (---) expt (ref 22).

Figure 8. Integral reaction cross section of reaction O+ + D2: () CC; (−·−) CS; (···) HD-RWP (ref 26); (---) expt (ref 22).

Figure 9a−c, we can observe that all the CC cross sections are within experimental error margins, nevertheless the CS ones are a bit lower than the experimental data at the lower collision energies. When paying the attention to the differences between the CC and CS cross sections of the global reaction O+ + HD → OH+ (OD+) + D(H), we find that the CS approximation apparently underestimates the cross sections at ET < 0.6 eV, and slightly overestimates the cross sections at ET > 0.6 eV, which are absolutely different from the results of O+ + H2/D2 reactions. The slight oscillatory character can also be observed. We also plot the integral cross sections of O+ + HD → OH+ + D and O+ + HD → OD+ + H reactions in logarithmic scales, as Figure S17 (Supporting Information) shows. We find that the CC cross section of the OH+ + D channel has a slope change at

the distribution of reaction probabilities in Figures S14 and S15 (Supporting Information). In Figure S16 (Supporting Information), we present the J-dependent partial wave contributions to the integral cross section of both channels at four collision energies. There are some differences between both channels. The partial wave contributions of OH+ + D channel at a different collision energy have sharp peaks at larger J, whereas for another channel, the partial wave contributions have broad peaks at moderate J. In addition, the contributions of OH+ + D channel is larger than that of OD+ + H channel. The integral cross sections of O+ + HD → OH+ (OD+) + D(H), O+ + HD → OH+ + D and O+ + HD → OD+ + H 10886

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which results from the Coriolis coupling effects on two product channels OH+ + D and OD+ + H. In light of the cross sections in Figure 9b,c, the CC cross sections are conspicuously larger than CS values in the OH+ + D channel at ET < 0.6 eV, whereas in the OD+ + H channel, the CC cross sections are a bit larger than CS values merely at ET < 0.3 eV, suggesting that Coriolis coupling effects influence the OH+ + D product channel much more than for the OD+ + H. This impression is further reinforced for the f(OH+) fraction in Figure 10, where the CC values is larger than CS ones at ET < 0.7 eV.

0.5 eV, which is larger than that of experimental value (0.4 eV). However, we cannot observe the slope change in the CC cross sections of the OD+ + H channel. The intramolecular isotopic effect of O+ + HD reaction expressed in terms of the f(OH+) fraction is presented in Figure 10. In the expression f(OH)+ = σ(OH)+/(σ(OH)+ + σ(OD)+),

IV. CONCLUSIONS We have carried out a time-dependent wave packet study on the O+ + H2 (vi = 0, ji = 0) reaction and its isotopic variants by ́ employing an accurate potential energy surface of Martinez et al. The calculated CC and CS probabilities and cross sections for the O+ + H2 reaction are essentially coincident with CCH and HD-RWP results, respectively, as expected. Through the comparison of the differences between the CC and CS cross sections, we find that the CS approximation overestimates/ slightly underestimates the cross sections at high/low collision energy for the O+ + H2/D2 reaction, whereas the CS approximation apparently underestimates the CC ones at ET < 0.6 eV, and slightly overestimates the cross sections at ET > 0.6 eV for the O+ + HD reaction. Through the comprehensive analysis of Coriolis coupling effect on the title reaction, we conclude that abstraction mechanism dominates the O+ + H2/ D2 reaction, and the Coriolis coupling effects influence the OH+ + D product channel of O+ + HD reaction much more. Comparing to the experimental results, our CC and CS cross sections are within experimental error margins, as the previously reported CCH and HD-RWP ones. Moreover, the CC cross sections of the O+ + H2 reaction can well reproduce the slope change that occurs at 0.3 eV. The trend of the f(OH+) values of O+ + HD reaction in CC calculations is in good agreement with the experimental measurements. To sum up, the Coriolis coupling effect plays a rather significant role in the O+ + H2 reaction and its isotopic variants and should be included in the accurate quantum dynamical calculations.

Figure 10. Intramolecular isotopic effect of the O+ + HD reaction expressed in terms of f(OH+) fraction: () CC; (−·−) CS; (···) HDRWP (ref 26); (---) expt (ref 22).

σ(OH+) and σ(OD+) denote the integral reaction cross section of reactions O+ + HD → OH+ + D and O+ + HD → OD+ + H, respectively. From the figure it can be clearly seen that both the CS and HD-RWP methods show an increase of the f(OH+)values from the lowest ET followed by a maximum value (0.7 at around ET = 0.25 eV) and then a slight decline. However, the behavior of f(OH+) fraction with respect to collision energy in CC calculations is rather different, ET > 0.6 eV. The f(OH+) value in CC calculations increases from the lowest ET and reaches the highest value at around ET = 0.4 eV. From this energy, the fraction gradually decreases. The overall behavior of the CC results fits well with the experimental f(OH+), even though the f(OH+) fraction in CC calculations is a larger than experimental measurements at ET < 0.7 eV. Through the comprehensive analysis of the CC and CS reaction probabilities and integral reaction cross sections of the O+ + H2 reaction and its isotopic variants, the marked differences between the two sets of calculations have proved the significant influence of Coriolis coupling effect on the title ionmolecular reaction. Now we will present an analysis of the Coriolis coupling effect. It is very obvious that the presence of the long-range potential in the O+ + H2 reaction makes the collision process take longer with reagent orientation, leading to severe deviation from the sudden collision and resulting in ́ pronounced Coriolis coupling effect. In addition, Martinez et al.24 identified the trajectories using the QCT method and proved that direct abstraction mechanism plays an important role for O+ + H2 reaction and its isotopic variants. As we all know, neglecting the Coriolis coupling effect in the CS approximation confines the molecular rotation to the molecular plane. According to the conclusions of Chu et al.,1,11 the confinement to the in-plane rotation increases the possibility for carrying the atom from the rotating molecule for an abstraction mechanism, especially at high collision energy, which is the main reason for the overestimation of the CS approximation in the dominant abstraction mechanism of the O+ + H2/D2 reaction. For the O+ + HD reaction, the CS cross sections clearly underestimate the CC results at collision energy ET < 0.5 eV,

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ASSOCIATED CONTENT

* Supporting Information S

The reaction probability dependence with collision energy, the distribution of reaction probability, the weighed partial wave contributions to the integral cross sections, and the logarithmic representation of the cross section for reactions O+ + D2 and O+ + HD. This information is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Prof. Miguel González and Dr. Rodrigo ́ Martinez for providing the potential energy surface and their theoretical results for comparisons. 10887

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