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Quantum dynamics study on the exchange H + OH+ reaction
Computational and Theoretical Chemistry 1012 (2013) 1–7
Contents lists available at SciVerse ScienceDirect
Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
Quantum dynamics study on the exchange H + OH+ reaction Wenwu Xu, Wenliang Li, Peiyu Zhang ⇑ State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 28 October 2012 Received in revised form 11 February 2013 Accepted 12 February 2013 Available online 6 March 2013 Keywords: Quantum Dynamics Wavepacket Potential Surface CS
a b s t r a c t The time-dependent wave packet quantum method under centrifugal sudden (CS) approximation has been employed to investigate the dynamics of the exchange reaction H + OH+ based on an accurate potential energy surface [Martínez et al., J. Chem. Phys. 120 (2004) 4705]. The reaction probability dependence with collision energy, the weighed partial wave contributions to the integral cross sections, and the integral cross sections of the exchange reaction H + OH+ in the collision energy range of 0.0–1.0 eV with reactant OH+ in the rotational state ji = 0 and vibrational states vi = 0–4 are calculated. The calculated time evolution of CS probability density distribution in logarithmic scales at total angular momentum J = 0 clearly indicates that the convex structure in the reaction path of the exchange H + OH+ reaction has significant influence on the dynamics of title reaction. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction During the development of ion–molecule reaction dynamics, the O+ + H2 reaction has played a great role. Over the years, the title reaction has attracted considerable attention both experimentally and theoretically [1–11]. On the theoretical side, Martínez et al. [3] developed an accurate analytical potential energy surface (PES) of Oþ ð4 SÞ þ H2 ðX 1 Rþ by fitting around 600 g Þ system CCSD(T)/cc-pVQZ ab initio point. Then the quasi-classical trajectory (QCT) [4], time-independent close-coupling hyperspherical (CCH) quantum dynamics [5] and time-dependent real wave packet quantum calculations [6] under the helicity decoupling (HDRWP) were performed by Martínez et al. to investigate the dynamics of the O+ + H2 reaction and its isotopic variants D2 and HD. The results demonstrated the good agreement between the theoretical calculations and experimental measurements. Recently, we have studied the influence of Coriolis coupling (CC) on the O+ + H2/D2/ HD reactions [7] because of the importance of the CC in the ion– molecule reaction dynamics [12–19]. Through the comparison between the results with and without CC, the pronounced CC effects have been revealed in the three reactions and should be included in the accurate quantum dynamical calculations. In a word, the O+ + H2 reaction and its isotopic variants have been investigated extensively. However, the H + OH+ reaction, which is the reverse reaction of O+ + H2, has been studied not much. Although the exchange H + OH+ reaction under study is too limited because of the difficulties to distinguish experimentally inelastic ⇑ Corresponding author. Tel.: +86 411 84379293. E-mail address: [email protected] (P. Zhang). 2210-271X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2013.02.006
and elastic collision from collision involving the exchange of the proton, theoretically there are some interesting phenomena can be found in the H + OH+ reaction. Fig. 1 shows the plot of the potential energy contour at rOHþ = 1.0281 Å in xy coordinates, where x = R cos c and yR sin c (R is the H–OH+ center of mass distance as shown in Fig. 1 and c is the Jacobi angle). The reaction path continues to the OHþ 2 bound state well as the H atom approaches OH+. The H + OH+ reaction has two product channels O+ + H2 and H + OH+, as shown in Fig. 2, which is obtained by scanning the PES. It is observed that there exists two potential wells. After entering into the first potential well, the reactants have two channels to leave the well. The first channel is that the reactants directly generate the product of O+ + H2 via a steep slope, which indicates that this product channel is unlikely to occur and is not worth to pay attention to. Concerning another channel, the reactants must pass over a convex structure to enter into the second potential well. Then they leave the well to generate the product of H + OH+. According to Fig. 2 we plot the schematic PES of the second channel in Fig. 3, namely the exchange H + OH+ ? H + OH+ reaction. We can clearly find that the potential well is divided by a convex structure into double symmetric wells with the same depth. Relative to the H + OH+ asymptote, the depth of the potential well is 0.395 eV. We can also find that the convex structure cannot be seen as a barrier, because the energy of the convex structure is 0.121 eV lower than that of the reactants (zero potential energy point). The double well structure is common in the chemical reaction dynamics, such as NOH [20] and COH [21] systems. The potential energy profile for both NOH and COH systems exhibit a double well structure with minima for HXO and HOX isomers (X denotes C and N). However, there
2
W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
Fig. 1. Potential energy contour diagram for the approach of a H atom toward OH+ in its equilibrium geometry. Energies (eV) are relative to H + OH+ asymptote.
exists essential difference between the exchange reaction H + OH+ and the XOH system, since the reaction path of the exchange H + OH+ reaction does not involve isomer. Here, the time-dependent quantum wave packet method under centrifugal sudden (CS) approximation is used to investigate the dynamics of the exchange H + OH+ reaction by employing an accurate potential energy surface of Martínez et al. The paper is organized as follows: in Section 2 the theoretical method and calculation details are described. The results and discussion are considered in Section 3. Finally, the conclusions are given in Section 4. 2. Time-dependent wave-packet method Quantum calculations are carried out using the time-dependent wave packet method developed by Zhang et al. [22,23] In the reactant Jacobi coordinates, the Hamiltonian of the studied system can be expressed as:
H¼
2 ^j2 @2 h ðbJ ^jÞ2 b ^r Þ þ hðrÞ þ þ þ Vð R; 2lR @R2 2lR R2 2lr r 2
Fig. 3. The schematic potential energy surface of the exchange reaction H + OH+. Energies (eV) are relative to H + OH+ asymptote.
where V(r) is the diatomic reference potential. The initial wave packet is expanded in terms of the body-fixed e b ^ (BF) translational–vibrational–rotational basis uvn ðRÞ/v ðrÞY JM jK ð R; r Þ, where n and v are the indices labeling the translational and vibrational eigenfunctions, respectively. M and K are the projection quantum numbers of J on the space-fixed and BF z-axis, respectively, and e is the parity of the system. Accordingly, the element of the centrifugal term in the CS approximation is expressed as: 2
h
2lR R2
D
E e b ^ 2 JMe Y JM ¼ jK jð J jÞ jY j0 K 0
h
2
2lR R2
djj0 ½JðJ þ 1Þ þ jðj þ 1Þ
2K 2 dKK 0
ð3Þ
b The time-dependent Schrödinger equation (i h @w ¼ Hw) is solv@t ing using the reactant Jacobi coordinates by the split-operator scheme as follows:
wðR; r; t þ DÞ ¼ eiH0 D=2 eiV rot D=2 eiV D=2 eiV rot D=2 eiH0 D=2 wðR; r; tÞ
ð4Þ
ð1Þ
where R is the distance from the H atom to the center-of-mass of OH+, r is the OH+ bond length, lR is the reduced mass of H with respect to OH+, lr is the reduced mass of OH+, J is the total angular momentum, j is the rotational angular momentum number of b ^r Þ is the interaction potential excluding the diatomic OH+, and Vð R; potential of OH+, and h(r) is the diatomic reference Hamiltonian 2
hðrÞ ¼
@2 h þ VðrÞ 2lr @r2
ð2Þ
Fig. 2. Potential energy contour diagram of H + OH+ reaction. Energies (eV) are relative to H + OH+ asymptote.
Fig. 4. Reaction probability dependence with collision energy ET of the exchange reaction H + OH+ (vi = 0, ji = 0) at total angular momentum J = 0, 5, 10 and 15.
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Fig. 5. Left panel: weighed partial wave contributions to the integral cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0) as a function of total angular momentum J at five collision energies 0.1, 0.4, 0.7 and 1.0 eV. Right panel: integral reaction cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0).
Fig. 6. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 0, ji = 0) reaction in logarithmic scales over the collision energy of 0.2–0.3 eV at J = 0. The width and average translation energy of initial wave packet are 3.5 a0 and 0.248 eV, respectively, which makes 99% of the initial wave packet is at collision energy of 0.2–0.3 eV. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
2
H0 ¼
V rot ¼
2
@2 h h @ 2 þ VðrÞ 2lR @R2 2lr @r 2
ðbJ ^jÞ2 2
2lR R
þ
^j2 2lr r 2
ð5Þ
ð6Þ
The total reaction probability is calculated from the total flux through a surface located in the product channel s = s0 as follows:
h @ PJv 0 j0 k0 ðEÞ ¼ Im wj ðEÞjdðs s0 Þ jwj ðEÞ @s lr
ð7Þ
Then the individual and total reaction cross sections can be calculated by:
rv 0 j0 j0 ðEÞ ¼
p 2
k
RJ ð2J þ 1ÞPJv 0 j0 k0 ðEÞ
ð8Þ
rv 0 j0 ðEÞ ¼
1 Rk rv j k ðEÞ 2j0 þ 1 0 0 0 0
ð9Þ
Table 1 Parameters for the quantum calculations of the H + OH+ reaction (all quantities are given in a.u., unless otherwise indicated). 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 jmax For the rotational basis functions Internal coordinate (r) range Propagation time Time step
14.0 0.461 0.3 0.1–24 320 150 130 0.5–18 90,000 10
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W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
Fig. 7. Reaction probability dependence with collision energy ET of the exchange reaction H + OH+ (vi = 0–4, ji = 0) at total angular momentum J = 0, 10, 20 and 30.
Fig. 8. Weighed partial wave contributions to the integral cross sections of the exchange reaction H + OH+ (vi = 0–4, ji = 0) as a function of total angular momentum J at two collision energies 0.7 and 1.0 eV.
In Eqs. (7)–(9), w(E) is the corresponding time-independent part of the final wave function, and k is the wave number corresponding to the initial state at a fixed collision energy E.
3. Results and discussion
Fig. 9. Integral reaction cross sections of the exchange reaction H + OH+ (vi = 0–4, ji = 0).
The J = 0, 5, 10, and 15 reaction probabilities of the exchange H + OH+ reaction at the initial vibrational and rotational quantum numbers vi = 0 and ji = 0 are presented in Fig. 4. As can be seen, the main feature of the probability for J = 0 is its dense oscillatory structure with a significant number of high intensity peaks in the low energy range. At higher energies, the peaks are of lower intensity. With the increased J, the density of oscillatory structure gradually becomes smaller. This oscillatory behavior can be attributed to the potential well of H + OH+ reaction. It is also clear that the
W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
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Fig. 10. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales over the collision energy of 0.2–0.4 eV at J = 0. The width and average translation energy of initial wave packet for the collision energy range of 0.2–0.4 eV are 1.9 a0 and 0.294 eV, respectively. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
Fig. 11. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales over the collision energy of 0.8–1.0 eV at J = 0. The width and average translation energy of initial wave packet for the collision energy range of 0.8–1.0 eV are 3.0 a0 and 0.898 eV, respectively. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
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shifting of the reaction threshold for large J is entirely due to the centrifugal barrier for the intrinsically barrierless reaction. In the left panel of Fig. 5, we present the J-dependent partial wave contributions (weighted over a 2J + 1 factor) to the integral cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0) at four collision energies of 0.1, 0.4, 0.7 and 1.0 eV. It can be seen from the figure that the contribution outspreads over a larger number of J with the increased collision energy. The right panel of Fig. 5 shows the integral reaction cross sections of the exchange H + OH+ (vi = 0, ji = 0) reaction. The WP propagation is performed for J = 0–25 in calculations. We find the integral reaction cross sections show no threshold and are very large at low collision energies, and they will decrease with the increased collision energy. Finally the cross sections reach a plateau. The trend of the curve is very similar to barrierless reactions O+ + H2 [3–6], He + HeH+ [24,25], D+ + H2 [26] as well as H + LiH+ [27]. In addition, strong oscillatory structure of the cross sections is also observed. Through the comparison of the integral reaction cross sections at the initial vibrational and rotational quantum numbers vi = 0 and ji = 0, we find that the integral reaction cross sections of the exchange H + OH+ reaction are far lower than that of the O+ + H2 reaction. In order to explain this phenomenon, we print the time evolution of CS probability density distribution in logarithmic scales over the collision energy of 0.2–0.3 eV at J = 0 in Fig. 6. The same parameters as Table 1 are used except the width and average translation energy of initial wave packet are 3.5 a0 and 0.248 eV, respectively, which makes 99% of the initial wave packet is at collision energy of 0.2–0.3 eV. At propagation time T = 1500 and 2500 au, we can see that the distribution gradually enters into the first potential well. While at T = 3000 and 3500 au, a rather small part of distribution can pass over the convex structure to enter into the second potential well, and the rest is blocked by the convex structure. When propagation time T = 4000 and 5000 au, the blocked distribution is rebounded by the convex structure. In a word, we can conclude from Fig. 6 that great number of probability density distribution, which is rebounded by the convex structure after entering into the first potential well, cannot arrive at the second potential well. This is the reason why the cross sections of the exchange H + OH+ reaction are much lower than that of the O+ + H2 reaction. Therefore, the convex structure plays a significant role in the dynamics of the exchange H + OH+ reaction. Usually, the oscillatory behavior of the reaction probabilities can be attributed to the potential well, and the cross sections have weak oscillation because of the washing out effects on the oscillatory probabilities by including different J values. Our previous calculations [7] on the O+ + H2 (vi = 0, ji = 0) reaction have similar characteristic in the oscillatory structure of reaction probabilities and cross sections. The WP propagation of the O+ + H2 reaction is performed for J = 0–80 in the calculations of the cross sections. However, strong oscillatory structure of the cross sections in the exchange H + OH+ reaction is observed, as shown in the right panel of Fig. 5. The WP propagation of the exchange H + OH+ reaction is performed for J = 0–25. But in the title reaction, there are a convex structure in the potential well, and the effective potential barrier is contributed not only the centrifugal energy but also the potential of convex structure. The enhanced effective potential barrier lead to much smaller number of J values needed in the exchange H + OH+ reaction than that in the O+ + H2 reaction to get converged cross sections. Thus the oscillatory structure in the reaction probabilities of the title reaction still exists in the cross sections. The effect of increasing the vibrational energy of the reactant on the probabilities for different J values is shown in Fig. 7. The probabilities for all J are almost unaffected by the increase in the vibrational energy up to vi = 2. While for higher vibrational quantum numbers vi = 3 and 4, the probabilities increase significantly with the increased collision energy. Fig. 8 presents the J-dependent par-
tial wave contributions to the integral cross sections of the exchange H + OH+ (vi = 0–4, ji = 0) reaction at two collision energies of 0.7 and 1.0 eV. Similar results to the reaction probabilities in Fig. 7 can also be found. It exists a low partial wave contribution at vi = 0, 1, and 2, and a substantial contribution at vi = 3 and 4, especially for 1.0 eV collision energy. We also plot the integral cross sections of the exchange reaction H + OH+ for different initial vibrational states and for ji = 0 in Fig. 9. The cross sections for all vibrational states show no threshold and are very large at low collision. With the increased collision energy, the cross sections for vi = 1 and 2 show a continuous decreasing trend and finally reach a plateau. While the vi = 3 and 4 cross sections values increase with the increased collision energy at high collision energy region. We plot the time evolution of reaction probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales at J = 0 over the collision energy of 0.2–0.4 eV in Fig. 10 and 0.8– 1.0 eV in Fig. 11, respectively. The width and average translation energy of initial wave packet for the collision energy range of 0.2–0.4 eV/0.8–1.0 eV are 1.9 a0 and 0.294 eV/3.0 a0 and 0.898 eV, respectively. It is clearly observed from the comparison between Figs. 10 and 11 that the probability density distribution for the collision energy range of 0.8–1.0 eV, which can pass over the convex structure to generate the product of H + OH+, is much larger than that for the 0.2–0.4 eV. Hence the convex structure has great influence on the distribution at high vibrational excitation and high collision energy. 4. Conclusion In this paper, a time-dependent quantum wave packet method under CS approximation was used to investigate the dynamics of the exchange reaction H + OH+ based on an accurate potential energy surface of Martínez et al. A convex structure is observed in the reaction path of the exchange reaction H + OH+. The calculated time evolution of CS probability density distribution in logarithmic scales at J = 0 clearly indicates that a majority of distribution cannot pass over the convex structure, leading to much less values of the cross sections in the exchange H + OH+ reaction than that in the O+ + H2 reaction. In addition, the convex structure has significant influence on the dynamical results at high collision energy when the vibrational energy of the reactant is increased up to vi = 3. Acknowledgements The authors thank Prof. Miguel González and Dr. Rodrigo Martínez for providing the potential energy surface. References [1] C.Y. Ng, State-selected and state-to-state ionmolecule reaction dynamics, J. Phys. Chem. A 106 (2002) 5953–5966. [2] M. González, M. Gilibert, A. Aguilar, R. Sayós, A comparison between experimental and theoretical excitation functions for the O+ + H2 (4A00 ) system using trajectory calculations over a wide energy range, J. Chem. Phys. 98 (1993) 2927–2935. [3] R. Martínez, J. Millán, M. González, Ab initio analytical potential energy surface and quasiclassical trajectory study of the þ 3 2 Oþ ð4 SÞ þ H2 ðX 1 Rþ g Þ ! OH ðX R Þ þ Hð SÞ reaction and isotopic variants, J. Chem. Phys. 120 (2004) 4705–4714. [4] R. Martínez, J.D. Sierra, M. González, Cross sections of the O+ + H2 ? OH+ + H ion–molecule reaction and isotopic variants (D2, HD): quasiclassical trajectory study and comparison with experiments, J. Chem. Phys. 123 (2005) 174312. [5] R. Martínez, J.M. Lucas, X. Giménez, A. Aguilar, M. González, Exact quantum dynamics study of the O+ + H2 (v = 0, j = 0) ? OH+ + H ion–molecule reaction and comparison with quasiclassical trajectory calculations, J. Chem. Phys. 124 (2006) 144301. [6] R. Martínez, J.D. Sierra, S.K. Gray, M. González, Time dependent quantum dynamics study of the O+ + H2 (v = 0, j = 0) ? OH+ + H ion–molecule reaction and isotopic variants (D2, HD), J. Chem. Phys. 125 (2006) 164305.
W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7 [7] W.W. Xu, W.L. Li, S.J. Lv, H.S. Zhai, Z.X. Duan, P.Y. Zhang, Coriolis coupling þ 3 2 effects in Oþ ð4 SÞ þ H2 ðX 1 Rþ g Þ ! OH ðX R Þ þ Hð SÞ reaction and its isotopic variants: exact time-dependent quantum scattering study, J. Phys. Chem. A 116 (2012) 10882–10888. [8] M. González, A. Aguilar, M. Gilibert, Dynamics of the ion–molecule reaction and some of its isotopic variants (D2 and HD). I. Potential energy surface and preliminary quasiclassical trajectory analysis, Chem. Phys. 131 (1989) 335–346. [9] M. González, A. Aguilar, M. Gilibert, Dynamics of the ion–molecule reaction and some of its isotopic variants (D2 and HD). II. Quasiclassical trajectory study in the range of relative energies 0.25–6.30 eV, Chem. Phys. 131 (1989) 347–364. [10] M. González, M. Gilibert, A. Aguilar, R. Sayós, Effect of reagent rotation on the dynamics of the O+ + H2 ion–molecule reaction and isotopic variants, Chem. Phys. Lett. 204 (1993) 578–586. [11] J.D. Burley, K.M. Ervin, P.B. Armentrout, Translational energy dependence of O+ (4S)+H2 (D2, HD) ? OH+ (OD+) + H(D) from thermal energies to 30 eV cm., Int. J. Mass Spectrom. Ion Process. 80 (1987) 153–175. [12] T.S. Chu, K.L. Han, Coriolis coupling effect in molecular reaction dynamics, Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 108 (2012) 10–33. [13] T.S. Chu, K.L. Han, Effect of coriolis coupling in chemical reaction dynamics, Phys. Chem. Chem. Phys. 10 (2008) 2431–2441. [14] T.S. Chu, Y. Zhang, K.L. Han, The time-dependent quantum wave packet approach to the electronically nonadiabatic processes in chemical reactions, Int. Rev. Phys. Chem. 25 (2006) 201–235. [15] Y. Zhang, T.X. Xie, K.L. Han, J.Z.H. Zhang, Time-dependent quantum wave packet calculation for nonadiabatic F(2P3/2, 2P1/2) + H2 reaction, J. Chem. Phys. 119 (2003) 12921–12925.
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[16] T.X. Xie, Y. Zhang, M.Y. Zhao, K.L. Han, Calculations of the F + HD reaction on three potential energy surfaces, Phys. Chem. Chem. Phys. 5 (2003) 2034–2038. [17] T.S. Chu, K.L. Han, G.C. Schatz, Significant nonadiabatic effects in the S(1D) + HD reaction, J. Phys. Chem. A 111 (2007) 8286–8290. [18] A.J. Zhang, J.F. Jia, H.S. Wu, An investigation of electronic nonadiabaticity in the D + HBr and H + DBr reaction on new diabatic potential energy surfaces, Commun. Comput. Chem. 1 (2013) 40–51. [19] P.Y. Zhang, S.J. Lv, Five new ab initio potential energy surfaces for the O(3P, 1 D) + H2 system, Commun. Comput. Chem. 1 (2013) 63–71. [20] R. Guadagnini, G.C. Schatz, S.P. Walch, Global potential energy surfaces for the lowest 1A0 , 3A00 , and 1A00 states of HNO, J. Chem. Phys. 102 (1995) 774–783. [21] M. Jorfi, P. Honvault, State-to-state quantum dynamics calculations of the C + OH reaction on the second excited potential energy surface, J. Phys. Chem. A 115 (2011) 8791–8796. [22] J.Z.H. Zhang, Theory and application of quantum molecular dynamics, World Scientific, Singapore, 1999. [23] J.Z.H. Zhang, J.Q. Dai, W. Zhu, Development of accurate quantum dynamical methods for tetraatomic reactions, J. Phys. Chem. A. 101 (1997) 2746–2754. [24] A.N. Panda, N. Sathyamurthy, Bound and quasibound states of He2H+ and He2D+, J. Phys. Chem. A 107 (2003) 7125–7131. [25] S. Bhattacharya, A.N. Panda, Time-dependent quantum dynamics of the He + H+He reaction, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 085201. [26] T.S. Chu, K.L. Han, Nonadiabatic time-dependent wave packet study of the D+ + H2 reaction system, J. Phys. Chem. A 109 (2005) 2050–2056. [27] T.G. Yang, J.C. Yuan, D.H. Cheng, M.D. Chen, Quasi-classical trajectory study of the effects of reactant to-vibrational excitation on the H + LiH+ ? Li+ + H2 reaction, Commun. Comput. Chem. 1 (2013) 15–26.
Contents lists available at SciVerse ScienceDirect
Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
Quantum dynamics study on the exchange H + OH+ reaction Wenwu Xu, Wenliang Li, Peiyu Zhang ⇑ State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 28 October 2012 Received in revised form 11 February 2013 Accepted 12 February 2013 Available online 6 March 2013 Keywords: Quantum Dynamics Wavepacket Potential Surface CS
a b s t r a c t The time-dependent wave packet quantum method under centrifugal sudden (CS) approximation has been employed to investigate the dynamics of the exchange reaction H + OH+ based on an accurate potential energy surface [Martínez et al., J. Chem. Phys. 120 (2004) 4705]. The reaction probability dependence with collision energy, the weighed partial wave contributions to the integral cross sections, and the integral cross sections of the exchange reaction H + OH+ in the collision energy range of 0.0–1.0 eV with reactant OH+ in the rotational state ji = 0 and vibrational states vi = 0–4 are calculated. The calculated time evolution of CS probability density distribution in logarithmic scales at total angular momentum J = 0 clearly indicates that the convex structure in the reaction path of the exchange H + OH+ reaction has significant influence on the dynamics of title reaction. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction During the development of ion–molecule reaction dynamics, the O+ + H2 reaction has played a great role. Over the years, the title reaction has attracted considerable attention both experimentally and theoretically [1–11]. On the theoretical side, Martínez et al. [3] developed an accurate analytical potential energy surface (PES) of Oþ ð4 SÞ þ H2 ðX 1 Rþ by fitting around 600 g Þ system CCSD(T)/cc-pVQZ ab initio point. Then the quasi-classical trajectory (QCT) [4], time-independent close-coupling hyperspherical (CCH) quantum dynamics [5] and time-dependent real wave packet quantum calculations [6] under the helicity decoupling (HDRWP) were performed by Martínez et al. to investigate the dynamics of the O+ + H2 reaction and its isotopic variants D2 and HD. The results demonstrated the good agreement between the theoretical calculations and experimental measurements. Recently, we have studied the influence of Coriolis coupling (CC) on the O+ + H2/D2/ HD reactions [7] because of the importance of the CC in the ion– molecule reaction dynamics [12–19]. Through the comparison between the results with and without CC, the pronounced CC effects have been revealed in the three reactions and should be included in the accurate quantum dynamical calculations. In a word, the O+ + H2 reaction and its isotopic variants have been investigated extensively. However, the H + OH+ reaction, which is the reverse reaction of O+ + H2, has been studied not much. Although the exchange H + OH+ reaction under study is too limited because of the difficulties to distinguish experimentally inelastic ⇑ Corresponding author. Tel.: +86 411 84379293. E-mail address: [email protected] (P. Zhang). 2210-271X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2013.02.006
and elastic collision from collision involving the exchange of the proton, theoretically there are some interesting phenomena can be found in the H + OH+ reaction. Fig. 1 shows the plot of the potential energy contour at rOHþ = 1.0281 Å in xy coordinates, where x = R cos c and yR sin c (R is the H–OH+ center of mass distance as shown in Fig. 1 and c is the Jacobi angle). The reaction path continues to the OHþ 2 bound state well as the H atom approaches OH+. The H + OH+ reaction has two product channels O+ + H2 and H + OH+, as shown in Fig. 2, which is obtained by scanning the PES. It is observed that there exists two potential wells. After entering into the first potential well, the reactants have two channels to leave the well. The first channel is that the reactants directly generate the product of O+ + H2 via a steep slope, which indicates that this product channel is unlikely to occur and is not worth to pay attention to. Concerning another channel, the reactants must pass over a convex structure to enter into the second potential well. Then they leave the well to generate the product of H + OH+. According to Fig. 2 we plot the schematic PES of the second channel in Fig. 3, namely the exchange H + OH+ ? H + OH+ reaction. We can clearly find that the potential well is divided by a convex structure into double symmetric wells with the same depth. Relative to the H + OH+ asymptote, the depth of the potential well is 0.395 eV. We can also find that the convex structure cannot be seen as a barrier, because the energy of the convex structure is 0.121 eV lower than that of the reactants (zero potential energy point). The double well structure is common in the chemical reaction dynamics, such as NOH [20] and COH [21] systems. The potential energy profile for both NOH and COH systems exhibit a double well structure with minima for HXO and HOX isomers (X denotes C and N). However, there
2
W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
Fig. 1. Potential energy contour diagram for the approach of a H atom toward OH+ in its equilibrium geometry. Energies (eV) are relative to H + OH+ asymptote.
exists essential difference between the exchange reaction H + OH+ and the XOH system, since the reaction path of the exchange H + OH+ reaction does not involve isomer. Here, the time-dependent quantum wave packet method under centrifugal sudden (CS) approximation is used to investigate the dynamics of the exchange H + OH+ reaction by employing an accurate potential energy surface of Martínez et al. The paper is organized as follows: in Section 2 the theoretical method and calculation details are described. The results and discussion are considered in Section 3. Finally, the conclusions are given in Section 4. 2. Time-dependent wave-packet method Quantum calculations are carried out using the time-dependent wave packet method developed by Zhang et al. [22,23] In the reactant Jacobi coordinates, the Hamiltonian of the studied system can be expressed as:
H¼
2 ^j2 @2 h ðbJ ^jÞ2 b ^r Þ þ hðrÞ þ þ þ Vð R; 2lR @R2 2lR R2 2lr r 2
Fig. 3. The schematic potential energy surface of the exchange reaction H + OH+. Energies (eV) are relative to H + OH+ asymptote.
where V(r) is the diatomic reference potential. The initial wave packet is expanded in terms of the body-fixed e b ^ (BF) translational–vibrational–rotational basis uvn ðRÞ/v ðrÞY JM jK ð R; r Þ, where n and v are the indices labeling the translational and vibrational eigenfunctions, respectively. M and K are the projection quantum numbers of J on the space-fixed and BF z-axis, respectively, and e is the parity of the system. Accordingly, the element of the centrifugal term in the CS approximation is expressed as: 2
h
2lR R2
D
E e b ^ 2 JMe Y JM ¼ jK jð J jÞ jY j0 K 0
h
2
2lR R2
djj0 ½JðJ þ 1Þ þ jðj þ 1Þ
2K 2 dKK 0
ð3Þ
b The time-dependent Schrödinger equation (i h @w ¼ Hw) is solv@t ing using the reactant Jacobi coordinates by the split-operator scheme as follows:
wðR; r; t þ DÞ ¼ eiH0 D=2 eiV rot D=2 eiV D=2 eiV rot D=2 eiH0 D=2 wðR; r; tÞ
ð4Þ
ð1Þ
where R is the distance from the H atom to the center-of-mass of OH+, r is the OH+ bond length, lR is the reduced mass of H with respect to OH+, lr is the reduced mass of OH+, J is the total angular momentum, j is the rotational angular momentum number of b ^r Þ is the interaction potential excluding the diatomic OH+, and Vð R; potential of OH+, and h(r) is the diatomic reference Hamiltonian 2
hðrÞ ¼
@2 h þ VðrÞ 2lr @r2
ð2Þ
Fig. 2. Potential energy contour diagram of H + OH+ reaction. Energies (eV) are relative to H + OH+ asymptote.
Fig. 4. Reaction probability dependence with collision energy ET of the exchange reaction H + OH+ (vi = 0, ji = 0) at total angular momentum J = 0, 5, 10 and 15.
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W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
Fig. 5. Left panel: weighed partial wave contributions to the integral cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0) as a function of total angular momentum J at five collision energies 0.1, 0.4, 0.7 and 1.0 eV. Right panel: integral reaction cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0).
Fig. 6. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 0, ji = 0) reaction in logarithmic scales over the collision energy of 0.2–0.3 eV at J = 0. The width and average translation energy of initial wave packet are 3.5 a0 and 0.248 eV, respectively, which makes 99% of the initial wave packet is at collision energy of 0.2–0.3 eV. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
2
H0 ¼
V rot ¼
2
@2 h h @ 2 þ VðrÞ 2lR @R2 2lr @r 2
ðbJ ^jÞ2 2
2lR R
þ
^j2 2lr r 2
ð5Þ
ð6Þ
The total reaction probability is calculated from the total flux through a surface located in the product channel s = s0 as follows:
h @ PJv 0 j0 k0 ðEÞ ¼ Im wj ðEÞjdðs s0 Þ jwj ðEÞ @s lr
ð7Þ
Then the individual and total reaction cross sections can be calculated by:
rv 0 j0 j0 ðEÞ ¼
p 2
k
RJ ð2J þ 1ÞPJv 0 j0 k0 ðEÞ
ð8Þ
rv 0 j0 ðEÞ ¼
1 Rk rv j k ðEÞ 2j0 þ 1 0 0 0 0
ð9Þ
Table 1 Parameters for the quantum calculations of the H + OH+ reaction (all quantities are given in a.u., unless otherwise indicated). 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 jmax For the rotational basis functions Internal coordinate (r) range Propagation time Time step
14.0 0.461 0.3 0.1–24 320 150 130 0.5–18 90,000 10
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W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
Fig. 7. Reaction probability dependence with collision energy ET of the exchange reaction H + OH+ (vi = 0–4, ji = 0) at total angular momentum J = 0, 10, 20 and 30.
Fig. 8. Weighed partial wave contributions to the integral cross sections of the exchange reaction H + OH+ (vi = 0–4, ji = 0) as a function of total angular momentum J at two collision energies 0.7 and 1.0 eV.
In Eqs. (7)–(9), w(E) is the corresponding time-independent part of the final wave function, and k is the wave number corresponding to the initial state at a fixed collision energy E.
3. Results and discussion
Fig. 9. Integral reaction cross sections of the exchange reaction H + OH+ (vi = 0–4, ji = 0).
The J = 0, 5, 10, and 15 reaction probabilities of the exchange H + OH+ reaction at the initial vibrational and rotational quantum numbers vi = 0 and ji = 0 are presented in Fig. 4. As can be seen, the main feature of the probability for J = 0 is its dense oscillatory structure with a significant number of high intensity peaks in the low energy range. At higher energies, the peaks are of lower intensity. With the increased J, the density of oscillatory structure gradually becomes smaller. This oscillatory behavior can be attributed to the potential well of H + OH+ reaction. It is also clear that the
W. Xu et al. / Computational and Theoretical Chemistry 1012 (2013) 1–7
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Fig. 10. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales over the collision energy of 0.2–0.4 eV at J = 0. The width and average translation energy of initial wave packet for the collision energy range of 0.2–0.4 eV are 1.9 a0 and 0.294 eV, respectively. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
Fig. 11. The time evolution of CS probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales over the collision energy of 0.8–1.0 eV at J = 0. The width and average translation energy of initial wave packet for the collision energy range of 0.8–1.0 eV are 3.0 a0 and 0.898 eV, respectively. The horizontal and vertical axes denote R (the distance of H–OH+) and r (the bond length of OH+), respectively.
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shifting of the reaction threshold for large J is entirely due to the centrifugal barrier for the intrinsically barrierless reaction. In the left panel of Fig. 5, we present the J-dependent partial wave contributions (weighted over a 2J + 1 factor) to the integral cross sections of the exchange reaction H + OH+ (vi = 0, ji = 0) at four collision energies of 0.1, 0.4, 0.7 and 1.0 eV. It can be seen from the figure that the contribution outspreads over a larger number of J with the increased collision energy. The right panel of Fig. 5 shows the integral reaction cross sections of the exchange H + OH+ (vi = 0, ji = 0) reaction. The WP propagation is performed for J = 0–25 in calculations. We find the integral reaction cross sections show no threshold and are very large at low collision energies, and they will decrease with the increased collision energy. Finally the cross sections reach a plateau. The trend of the curve is very similar to barrierless reactions O+ + H2 [3–6], He + HeH+ [24,25], D+ + H2 [26] as well as H + LiH+ [27]. In addition, strong oscillatory structure of the cross sections is also observed. Through the comparison of the integral reaction cross sections at the initial vibrational and rotational quantum numbers vi = 0 and ji = 0, we find that the integral reaction cross sections of the exchange H + OH+ reaction are far lower than that of the O+ + H2 reaction. In order to explain this phenomenon, we print the time evolution of CS probability density distribution in logarithmic scales over the collision energy of 0.2–0.3 eV at J = 0 in Fig. 6. The same parameters as Table 1 are used except the width and average translation energy of initial wave packet are 3.5 a0 and 0.248 eV, respectively, which makes 99% of the initial wave packet is at collision energy of 0.2–0.3 eV. At propagation time T = 1500 and 2500 au, we can see that the distribution gradually enters into the first potential well. While at T = 3000 and 3500 au, a rather small part of distribution can pass over the convex structure to enter into the second potential well, and the rest is blocked by the convex structure. When propagation time T = 4000 and 5000 au, the blocked distribution is rebounded by the convex structure. In a word, we can conclude from Fig. 6 that great number of probability density distribution, which is rebounded by the convex structure after entering into the first potential well, cannot arrive at the second potential well. This is the reason why the cross sections of the exchange H + OH+ reaction are much lower than that of the O+ + H2 reaction. Therefore, the convex structure plays a significant role in the dynamics of the exchange H + OH+ reaction. Usually, the oscillatory behavior of the reaction probabilities can be attributed to the potential well, and the cross sections have weak oscillation because of the washing out effects on the oscillatory probabilities by including different J values. Our previous calculations [7] on the O+ + H2 (vi = 0, ji = 0) reaction have similar characteristic in the oscillatory structure of reaction probabilities and cross sections. The WP propagation of the O+ + H2 reaction is performed for J = 0–80 in the calculations of the cross sections. However, strong oscillatory structure of the cross sections in the exchange H + OH+ reaction is observed, as shown in the right panel of Fig. 5. The WP propagation of the exchange H + OH+ reaction is performed for J = 0–25. But in the title reaction, there are a convex structure in the potential well, and the effective potential barrier is contributed not only the centrifugal energy but also the potential of convex structure. The enhanced effective potential barrier lead to much smaller number of J values needed in the exchange H + OH+ reaction than that in the O+ + H2 reaction to get converged cross sections. Thus the oscillatory structure in the reaction probabilities of the title reaction still exists in the cross sections. The effect of increasing the vibrational energy of the reactant on the probabilities for different J values is shown in Fig. 7. The probabilities for all J are almost unaffected by the increase in the vibrational energy up to vi = 2. While for higher vibrational quantum numbers vi = 3 and 4, the probabilities increase significantly with the increased collision energy. Fig. 8 presents the J-dependent par-
tial wave contributions to the integral cross sections of the exchange H + OH+ (vi = 0–4, ji = 0) reaction at two collision energies of 0.7 and 1.0 eV. Similar results to the reaction probabilities in Fig. 7 can also be found. It exists a low partial wave contribution at vi = 0, 1, and 2, and a substantial contribution at vi = 3 and 4, especially for 1.0 eV collision energy. We also plot the integral cross sections of the exchange reaction H + OH+ for different initial vibrational states and for ji = 0 in Fig. 9. The cross sections for all vibrational states show no threshold and are very large at low collision. With the increased collision energy, the cross sections for vi = 1 and 2 show a continuous decreasing trend and finally reach a plateau. While the vi = 3 and 4 cross sections values increase with the increased collision energy at high collision energy region. We plot the time evolution of reaction probability density distribution of the exchange H + OH+ (vi = 4, ji = 0) reaction in logarithmic scales at J = 0 over the collision energy of 0.2–0.4 eV in Fig. 10 and 0.8– 1.0 eV in Fig. 11, respectively. The width and average translation energy of initial wave packet for the collision energy range of 0.2–0.4 eV/0.8–1.0 eV are 1.9 a0 and 0.294 eV/3.0 a0 and 0.898 eV, respectively. It is clearly observed from the comparison between Figs. 10 and 11 that the probability density distribution for the collision energy range of 0.8–1.0 eV, which can pass over the convex structure to generate the product of H + OH+, is much larger than that for the 0.2–0.4 eV. Hence the convex structure has great influence on the distribution at high vibrational excitation and high collision energy. 4. Conclusion In this paper, a time-dependent quantum wave packet method under CS approximation was used to investigate the dynamics of the exchange reaction H + OH+ based on an accurate potential energy surface of Martínez et al. A convex structure is observed in the reaction path of the exchange reaction H + OH+. The calculated time evolution of CS probability density distribution in logarithmic scales at J = 0 clearly indicates that a majority of distribution cannot pass over the convex structure, leading to much less values of the cross sections in the exchange H + OH+ reaction than that in the O+ + H2 reaction. In addition, the convex structure has significant influence on the dynamical results at high collision energy when the vibrational energy of the reactant is increased up to vi = 3. Acknowledgements The authors thank Prof. Miguel González and Dr. Rodrigo Martínez for providing the potential energy surface. References [1] C.Y. Ng, State-selected and state-to-state ionmolecule reaction dynamics, J. Phys. Chem. A 106 (2002) 5953–5966. [2] M. González, M. Gilibert, A. Aguilar, R. Sayós, A comparison between experimental and theoretical excitation functions for the O+ + H2 (4A00 ) system using trajectory calculations over a wide energy range, J. Chem. Phys. 98 (1993) 2927–2935. [3] R. Martínez, J. Millán, M. González, Ab initio analytical potential energy surface and quasiclassical trajectory study of the þ 3 2 Oþ ð4 SÞ þ H2 ðX 1 Rþ g Þ ! OH ðX R Þ þ Hð SÞ reaction and isotopic variants, J. Chem. Phys. 120 (2004) 4705–4714. [4] R. Martínez, J.D. Sierra, M. González, Cross sections of the O+ + H2 ? OH+ + H ion–molecule reaction and isotopic variants (D2, HD): quasiclassical trajectory study and comparison with experiments, J. Chem. Phys. 123 (2005) 174312. [5] R. Martínez, J.M. Lucas, X. Giménez, A. Aguilar, M. González, Exact quantum dynamics study of the O+ + H2 (v = 0, j = 0) ? OH+ + H ion–molecule reaction and comparison with quasiclassical trajectory calculations, J. Chem. Phys. 124 (2006) 144301. [6] R. Martínez, J.D. Sierra, S.K. Gray, M. González, Time dependent quantum dynamics study of the O+ + H2 (v = 0, j = 0) ? OH+ + H ion–molecule reaction and isotopic variants (D2, HD), J. Chem. Phys. 125 (2006) 164305.
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