Time-Dependent Wave-Packet Quantum Dynamics Study of the Ne +

Article pubs.acs.org/JPCA

Time-Dependent Wave-Packet Quantum Dynamics Study of the Ne + D2+ (v0 = 0−2, j0 = 0) → NeD+ + D Reaction: Including the Coriolis Coupling Cui-Xia Yao†,‡ and Pei-Yu Zhang*,† †

State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, 457 Zhongshan Road, Dalian 116023, China ‡ University of Chinese Academy of Sciences, 19A Yuquanlu, Beijing 100049, China ABSTRACT: The dynamics of the Ne + D2+ (v0 = 0−2, j0 = 0) → NeD+ + D reaction has been investigated in detail by using an accurate time-dependent wave-packet method on the ground 12A′ potential energy surface. Comparisons between the Coriolis coupling results and the centrifugal-sudden ones reveal that Coriolis coupling effect can influence reaction dynamics of the NeD2+ system. Integral cross sections have been evaluated for the Ne + D2+ reaction and its isotopic variant Ne + H2+, and a considerable intermolecular isotopic effect has been found. Also obvious is the great enhancement of the reactivity due to the reagent vibrational excitation. Besides, a comparison with previous theoretical results is also presented and discussed.

I. INTRODUCTION Ion−molecule reactions have a relevant role in interstellar chemistry, planetary ionospheres, and so on.1−3 Processes involving the NeH2+ system have been intensively investigated in the last 30 years, as this system is also of great importance for practical purposes, mainly for plasmas physics. For example, because Ne atoms can efficiently eliminate the excited H2+ ions, adding Ne atoms to the hydrogen plasma system can cool the high-temperature plasma close to the walls. Moreover, the reaction Ne + H2+→ NeH+ + H can be regarded as a prototype of moderately endothermic ion−molecular reactions. It takes place via the ground potential energy surface (PES), 12A′.4−11 This ground PES has similar characteristics to the HeH2+ system,12−14 with a collinear [Ne−H−H]+ well placed along the minimum energy path (MEP). The reduced number of electrons and the involvement of single PES make it attractive to experimentalist and theoreticians. Much effort has been done to construct the PES for the 12A′ state.15−22 The most widely used PHHJ-3 PES was reported by Pendergast et al. in 1993.19 With this surface, quasiclassical trajectory (QCT) and quantum dynamics studies, using both time-dependent (TD) and time-independent (TI) methods, are involved to study the Ne + H2+ system.23−30 In 2010, Lv et al. presented a new and more precise LZHH PES for the 12A′ state based on multireference configuration interaction (MRCI) ab initio calculations.31 This PES shows a collinear [Ne−H− H]+ well, which corresponds to the global minimum of the NeH2+ system. Departure from the collinear geometry will lead to a gradually higher barrier to this system. The root-meansquare (rms) error (0.27 kcal/mol) and the maximum energy © 2014 American Chemical Society

deviation (2.17 kcal/mol) are smaller, and the well depth on this surface is ∼0.019 eV deeper than those of the PHHJ-3 one.19 The schematic reaction path and energetics for the Ne + H2+ reaction on the LZHH PES have been shown in Figure 1.

Figure 1. Schematic reaction path and energetics for the Ne + H2+ reaction on the LZHH PES. (Distances are in bohr, and energies are in electronvolts.)

This ground PES is characterized by a well 0.54 eV below the Ne + H2+ asymptote. The endothermicity is 0.54 eV when the zero-point energies of the reactants H2+ and products NeH+ are considered for v0 = 0. More comparisons with the previous PHHJ-3 PES can be found in figure 2 in ref 31. Using this recent LZHH PES, Gamallo and coworkers investigated the Received: April 11, 2014 Revised: June 19, 2014 Published: June 20, 2014 5076

dx.doi.org/10.1021/jp503560j | J. Phys. Chem. A 2014, 118, 5076−5082

The Journal of Physical Chemistry A

Article

1 ε JMε ̂ ⟨YjK |(J − j ̂)2 |YjJM ′K′⟩ 2μR R2 1 δjj ′{[J(J + 1) + j(j + 1) − 2K 2]δKK ′ = 2μR R2

oscillations that were found in the integral cross section of the Ne + H2+ reaction, including both TD and TI quantum calculations.32,33 They analyzed that the oscillations could be attributed to the influence of the collinear [Ne−H−H]+ minimum on dynamics, which probably corresponds to Feshbach resonances. The isotopic effect on the stereodynamic properties was also conducted by running QCT trajectories.34−39 In the present work, we employ the accurate time-dependent wave-packet (TDWP) method to study the endoergic reaction Ne + D2+ (v0 = 0−2, j0 = 0) → NeD+ + D on the LZHH PES.31 With the purpose of analyzing the role of Coriolis coupling (CC) effect, we also perform the centrifugal sudden (CS) approximation in the calculations, which ignores Coriolis coupling. Furthermore, we discuss the intermolecular isotope effect and the influence of initial vibrational excitation on the dynamics. This contribution is organized as follows. The computational method is described in Section II, and in Section III the results and discussions are given. Finally, Section IV concludes.

+ + − − λjK (1 + δK 0)1/2 δK + 1, K ′ − λJK λjK (1 + δK1)1/2 δK − 1, K ′} − λJK

(4)

with ± λAB = [A(A + 1) − B(B ± 1)]1/2

For the case of CC calculations, it is well known that different K states couple to each other through the centrifugal potential for J > 0. The off-diagonal elements of the matrix are constituted. Usually, CC calculations are computationally expensive. In the case of CS calculations, we can neglect such off-diagonal elements, and the matrix becomes diagonal for simplicity. 1 JMε ε ⟨YjK |(J ̂ − j ̂)2 |YjJM ′K′ ⟩ 2μR R2 1 = δjj ′{[J(J + 1) + j(j + 1) − 2K 2]δKK ′} 2μR R2

The TDWP method used in this work has been described in detail by Zhang40 and applied successfully to a variety of collision reactions.41−48 Therefore, only a brief description concerning the title reaction will be shown here. For this triatomic reaction Ne + D2+, the Schrodinger equation is written as (ℏ = 1)

Ψ(t + Δ) = e−iH0Δ /2e−iVrotΔ /2e−iV Δe−iVrotΔ /2e−iH0Δ /2Ψ(t ) (7)

∂ i Ψ = Ĥ Ψ ∂t

2

(1)

H0 = −

Reactant Jacobi coordinates (R, r, γ) are chosen to describe the Hamiltonian42

Vrot =

∑ nvjK

ĵ

2μr r 2

+

(8)

(J ̂ − j ̂)2 2μR R2

(9)

After a sufficient long time of propagation, we can extract useful quantum attributes such as the total reaction probability and the integral cross section40

where R is the distance from Ne to the center of D2+ molecule, r describes the bond length of the reactant D2+, and γ acts as the enclosed angle. μR and μr are the reduced mass of NeD2+ system and D2+, respectively. J indicates the total angular momentum of this NeD2+ system, and j is the rotational angular momentum of D2+. V(R, r, γ) is the intermolecular potential, which in this case is the LZHH PES of Lv et al.31 The wave function which describes the nuclear motion can be written as47 JMε JMε ̂ FnvjKv (t )unv(R )φv (r )YjK (R ,r )̂ 0j0 k 0

2

1 ∂ 1 ∂ − + V (r ) 2 2μR ∂R 2μr ∂r 2 2

2 ĵ (J ̂ − j ̂)2 1 ∂2 1 ∂2 − + + + V (R , r , γ ) Ĥ = − 2μR ∂R2 2μr ∂r 2 2μr r 2 2μR R2 (2)

t) =

(6)

The computational time will be largely reduced when we use this CS approximation. However, it is not a common applicable rule to use this approximation for any collision system. Therefore, application of the CS approximation for computations must be pretested. The TD nuclear Schrodinger equation is solved numerically by the split-operator scheme on the LZHH PES.

II. CALCULATION DETAILS

ΨvJK0j εk 0(R̂ ,r ̂, 0

(5)

P Jj k 0v0(Ecoll) =

1 Im[ μr

σ(v0j k 0)(Ecoll) =

π k2

0

0

σv0j (Ecoll) = 0

(10)

∑ (2J + 1)PvJ j k (Ecoll) 00 0

J

1 2j0 + 1

(11)

∑ σv j k (Ecoll) 00 0

k0

(12)

Here Ψ(Ecoll) denotes the corresponding TI part of the final wave function. The best set of parameters is used to get the converged results. In our case, 200 translational basis functions for the R coordinate when R changes from 0.5 to 20.0 a0, 150 vibrational basis functions for the r in the range of 0.5−17.0 a0, and 100 rotational basis ones. Because a cationic system is considered to need a larger initial separation, the adequacy of the center of the initial wave packet Z0 has been reported in Figure 2, where no difference exists when Z0 changes from 13.5 bohr to 20.0 bohr. 80 partial waves are needed for the Ne + H2+ (v0 = 0, j0 =

(3)

̂ Here uvn(R), φv(r), and YJMε jK (R, r̂) denote the translational basis, the vibrational basis, and the rotational basis, respectively. v0, j0, and k0 are the indices labeling the initial rovibrational state of D2+. K is the projection of J along body-fixed (BF) z axes and ε is the parity of this NeD2+ system. On the basis of the rotational basis set, the centrifugal potential matrix can be shown as 5077



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III. RESULTS AND DISCUSSION Figure 4 presents the reaction probabilities as a function of collision energy for the Ne + D2+ (v0 = 0, j0 = 0) reaction for selected J values J = 10, 20, 30, 40, 50, and 60 using the TDWPCC and TDWP-CS methodologies. At low J values (J ≤ 30), the agreement between the CC and CS results is still very good, except for relative sharp oscillations in CS calculations. As J increases, the difference between the CC and CS results clearly increases: the CS reaction probabilities decrease more rapidly than those calculated from CC calculations. In the case of J = 60, this difference becomes enormous. Clearly, the CS approximation tends to underestimate the value of the reaction probability as J increases. This influence of Coriolis coupling on the Ne + D2+ reaction can be further supported by comparing the integral cross sections (ICSs) from the two calculations, as shown in Figure 5. A good accord is found for collision energies below ∼0.6 eV. In the higher collision energy, the trend of the lines and the values themselves is quite different for the two sets of calculations. The CC-ICSs increase sharply near the threshold, reach a maximum at the collision energy ∼1.0 eV, and then decline with further increasing collision energy (Ecoll). For the case of CS calculations, the cross sections rise quickly near the threshold and increase slightly with the collision energy when Ecoll > 0.6 eV. The TDWP-CC cross sections are obviously larger than the CS ones. The observation again confirms the conclusion of Chu et al.49 The existence of longrange attractive interactions of LZHH PES leads to the increase in the collision time and the slow rearrangement of atoms. Just like the Ne + D2+ reaction, the molecule rotating in all kinds of directions has the opportunity to break the bond of the collision complex NeD2+ to form the NeD+ product. In the case of CS approximation, the rotation of D2+ is limited by certain conditions,49,50 which leads to a poor role in escaping from the collision complex NeD2+. Because of the necessity of using the accurate quantum scattering calculations, the calculated TDWP-CC results are only presented and discussed in the following part. We compare the TDWP-CC excitation functions (dependence on the collision energy of the reactive integral cross section) between the Ne + H2+ (v0 = 0, j0 = 0) and Ne + D2+ (v0 = 0, j0 = 0) reactions in Figure 6a. Both of the results displayed here are computed in this work. It is apparent from the Figure that the trend of the lines is quite the same for both reactions; that is, excitation functions exhibit a quick increase at the threshold followed by a maximum value and then a slight decline with further increasing collision energy. However, the difference between the excitation functions is also noticeable: the resonances observed in the Ne + D2+ reaction are smoother than that obtained in the Ne + H2+ reaction and the values decrease significantly as the mass of the diatomic molecule increases. This behavior suggests an appreciable intermolecular isotope effect. The intermolecular isotopic effect can be seen even better in Figure 6b by taking into account the ratio of the excitation function for the two isotopic systems, for example, σ (H2+)/σ (D2+). Overall, the ratio is above 1 and tends to increase with Ecoll, somewhat accompanied by resonances in the low collision energy range, although the dependence is not monotonic. According to the discussion in classical terms,25 the maximum impact parameters (bmax) for both reactions are quite close to each other. Therefore, the origin of the observed differences in reactive cross sections mainly comes from the reaction probability.

Figure 2. Convergence tests of the center of the initial wavepacket for the J = 0 reaction probabilities as a function of collision energy.

0) reaction to achieve convergence up to the collision energy of 1.5 eV, while 104, 110, and 119 partial waves are necessary for the Ne + D2+ (v0 = 0−2, j0 = 0) reactions, respectively. We must consider the contribution from the higher projection of J (denoted as NK) onto the quantization axis. NK indicates the number of K states included in accurate CC calculations. For the NeD2+ system, we consider that the MEP from reactants Ne + D2+ to products NeD+ + D corresponds to the collinear or near collinear geometry. There is no need to include all projections K. The dependence of reaction probabilities on the number of K values at three selected J values (J = 15, 40, and 90) is presented in Figure 3a−c, respectively. From Figure 3a,b,

Figure 3. Dependence of reaction probabilities on NK included in Coriolis coupling calculations for the initial state v0 = 0, j0 = 0 and total angular momentum (a) J = 15. (b) J = 40, and (c) J = 90.

we can find that at low and intermediate J values the convergence of the probability is quick. In the case of the larger total angular momentum J = 90, as shown in Figure 3c, we can find that at the high energy region the probabilities still reduce when NK changes from 6 to 7, whereas there is little difference in probabilities between the case of NK = 7 and 8. The reaction probabilities are reasonably well-converged in the entire range of collision energies when NK = 7 is selected as the optimal value for accurate scattering calculations. The propagation time is set as 50 000 au. 5078



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Figure 4. Comparison between the CC and CS probabilities for the Ne + D2+ reaction in the collision energy of 0.5 to 1.5 eV for initial quantum numbers v0 = 0 and j0 = 0 and for total angular momenta J = 10, 20, 30, 40, 50, and 60.

As shown in Figure 7, the reaction probabilities from both reactions display oscillatory structure with Ecoll, which suggest numerous resonances. We can attribute the obviously observed peaks to the formation of metastable NeH2+ or NeD2+ states, which can be formed in the well of the LZHH PES. Apparent differences exist between the title reaction and its isotopic Ne + H2+ reaction. The reaction threshold of the Ne + D2+ reaction shifts to lower collision energy, and this trend becomes more conspicuous with increasing J. As can be expected, the reaction Ne + H2+ → NeH+ + H, ΔD00 = 0.54 eV, belongs to endoergic proton transfer reactions. The increased mass of D2+ results in the decrease in endoergicity for the Ne + D2+ → NeD+ + D, ΔD00 = 0.53 eV, reaction.32,51 Moreover, we can find that the probabilities for the Ne + D2+ reaction are smaller than that obtained for the Ne + H2+ reaction at high collision energies. This can be understood as follows. The lighter mass of H2+ molecule rotates faster than D2+, and thus the inertia momentum of the metastable NeH2+ is smaller. The lighter mass of H2+ can easily escape from the collision complex NeH2+, obtain favorable orientation geometry, and form the product NeH+. Similar phenomenon can also be observed for other reactions such as Li + H2+ and its isotopic variants,52 where a considerable intermolecular isotope effect was also found. Figure 8 displays the state-to-all reaction probabilities as a function of total energy (Etol = Ecoll + Ev0 (H2+)) for J = 0 for the reaction Ne + D2+ (v0 = 0−2, j0 = 0) and its isotopic variant Ne + H2 + (v0 = 0−2, j0 = 0), respectively. Corresponding results of the Ne + H2+ reaction in this Figure were calculated by Huarte-Larranaga et al. using the exact 3-D hyperspherical method on the PHHJ-3 PES.53 As can be seen, the reaction probabilities show a similar dense structure of sharp resonances for the isotopic variant H2+. However, the average width of the probability peaks for the Ne + H2+ system is somewhat denser than that for the related Ne + D2+ system. It is pointed out that the probability of formation and decay of metastable states depends on the isotopic variant: the lighter H allows a more efficient trapping and the lifetime of the complex NeH2+ state is longer than that of NeD2+. From Figure 9, we can get a better understanding of the influence of reagent vibrational excitation versus collision

Figure 5. Comparison between the TDWP-CC and TDWP-CS integral cross sections for the Ne + D2+ (v0 = 0, j0 = 0) → NeD+ + D reaction.

Figure 6. (a) Total integral reaction cross sections of the Ne + H2+ and Ne + D2+ reactions as a function of collision energy for the ground vibrotational state. (b) Ratio of the excitation function for the two isotopic systems σ (H2+)/σ (D2+) as a function of collision energy.

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Figure 7. TDWP-CC reaction probabilities as a function of collision energy at J = 0, 20, 30, 50, 60, and 70 for the Ne + H2+ (v0 = 0, j0 = 0) and Ne + D2+ (v0 = 0, j0 = 0) reactions.

Figure 9. Dependence of the integral cross section for the Ne + D2+ reaction on the total energy for the first three reactant D2+ vibrational levels. All cases correspond to initial j0 = 0. The dotted lines indicate Ecoll = 0.5, 0.7, 0.9, and 1.1 eV from left to right.

Figure 8. Reaction probabilities as a function of total energy at J = 0 for the Ne + H2+ (v0 = 0−2, j0 = 0) and Ne + D2+ (v0 = 0−2, j0 = 0) reactions. Note the scale change when going from v0 = 0 to v0 = 2. The results for the Ne + H2+ reaction are taken from ref 53.

the LZHH PES, and the barrier height corresponds to the reaction endoergicity 0.53 eV. These findings are in good agreement with Polanyi’s rules; that is, on “surface II”, where the barrier was in the exit valley of the PES, vibration was obviously more effective than translation to enhance the reactivity.54

energy, where the TDWP-CC integral cross sections are depicted as a function of total energy for the Ne + D2+ (v0 = 0− 2, j0 = 0) reaction. Dotted lines connect results at a particular collision energy with different vibrational states, whereas solid lines are the total cross sections for a given vibrational state. It is shown that not only are the probabilities influenced by numerous sharp resonances but even the cross sections are characterized by the resonance structure. The behavior of the ICS curves is analogous for three different vibrational states. They increase at low total energy and then decrease slightly. In addition, the cross sections for high v0 states have markedly larger values than those for low v0 states. They increase almost linearly with the vibrational energy of D2+ at a given collision energy. The vibrational excitation of the D2+ molecular is much more efficient than the translational energy in promoting reaction. This can be attributed to the late barrier character of

IV. CONCLUSIONS We have carried out quantum scattering studies of the Ne + D2+ (v0 = 0−2, j0 = 0) → NeD+ + D reaction in the collision energy of 0.5 to 1.5 eV on the new LZHH PES. To assess the influence of Coriolis coupling on the title reaction, the CS approximation that ignores CC is also employed in the quantum calculations for comparison purpose. It is found that one has to use the CC treatment to obtain quantitatively results for the title reaction. Despite the same trend of TDWP-CC cross sections for the two isotopic variants (H2+ and D2+), the increased mass of D2+ obviously reduces the incidence of reactivity taking place, predicting a large intermolecular isotope effect. In addition, integral cross sections are positively sensitive to the initial vibration state. This behavior is found to be pertinent to the late barrier character of the LZHH PES. 5080



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +86-411-84379381. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NSFC (11304310). REFERENCES

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