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Effects of rovibrational excitation of LiH on the LiH depletion and H exchange channels for the reaction H (2 S) + LiH (X1Σ+ ) on a new potential energy surface
Chemical Physics Letters 740 (2020) 137043
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Research paper
Effects of rovibrational excitation of LiH on the LiH depletion and H exchange channels for the reaction H ( 2S) + LiH (X1Σ+) on a new potential energy surface Xiaolin Wanga,b, Yujun Zhenga, Huan Yanga, a b
T
⁎
School of Physics, Shandong University, Jinan 250100, China School of Physics and Optoelectronic Engineering, Weifang University, Weifang 261061, China
H I GH L IG H T S
of QCT Reproduces the Results of QM Very Well. • Method of Rotational and Vibrational Excitations to the Reaction Dynamics are Studied. • Effects of Dynamics are Shown for Energies Below 0.06 eV. • Results • Totally Different Effects of Rotational Excitations are Found.
A R T I C LE I N FO
A B S T R A C T
Keywords: QCT Vibrational excitation Rotational excitation PES
The quasiclassical trajectory method is used to calculate the dynamics of the reaction H + LiH for both reaction channels. The initial state-selected and energy-resolved integral cross-sections, differential cross-sections, and state distributions of the products are investigated in the collision energy range 0.02–0.56 eV. Results show that both vibrational and rotational excitations reduce the reactivity of the LiH depletion (R1) channel but promote that of the H exchange (R2) channel. Forward scattering is prominent in the R1 channel, but backscattering is slightly favored in the R2 channel. The influence of vibrational excitation is reduced at higher energies.
1. Introduction Due to the importance of the LiH molecule and the H′ + LiH reaction in early cosmochemistry [1–9], considerable effort has been directed into understanding the reaction dynamics of this reaction system from both quantum-mechanical (QM) [7,10–13,8,14–24] and quasiclassical [25–29,15,30–32] perspectives. The H + LiH reaction proceeds through the highly exothermic LiH depletion (R1) and thermoneutral H exchange (R2) channels,
H′ + LiH → Li + H2
(R1)
H′ + LiH → H+ LiH′
(R2)
A prerequisite for performing such studies is an accurate potential energy surface (PES) [33,34]. Since 2009, the most extensively used global three-dimensional PES for the title reaction has been the PES of Wernli et al. [10] (‘Wernli PES’ hereafter) and that of Prudente et al. [24] (‘Prudente PES’ hereafter). The main difference between these
⁎
PESs is that the Prudente PES has an early-stage energy barrier, whereas the Wernli PES does not [11]. The main advantage of the Wernli PES is said to be a more accurate description of long-range forces between the fragments and the elimination of the barrier in the entrance channel [7]. Because of these differences, the physical pictures obtained from these two PESs are obviously different, especially at low collision energies [11]. Various QM calculations have been carried out on the Wernli PES. Roy and Mahapatra studied the effect of rovibrational excitations of the reactant in their time-dependent wave packet (TDWP) calculations within the centrifugal sudden approximation (CSA) [18]. However, due to the barrierless nature of the title reaction, the CSA procedure is not appropriate for dynamical calculations in this system [19,35]. He et al. performed GPU-based calculations that accounted for the vibrational quantum number of the reactant (v = 0, 1) and treated the Coriolis coupling (CC) terms accurately [19], but only the R1 channel was taken into account. In 2015, a new global PES (‘Chen PES’ hereafter) for the lowest
Corresponding author. E-mail address: [email protected] (H. Yang).
https://doi.org/10.1016/j.cplett.2019.137043 Received 27 October 2019; Received in revised form 11 December 2019; Accepted 13 December 2019 Available online 28 December 2019 0009-2614/ © 2019 Published by Elsevier B.V.
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doublet state of the LiH2 ( 2A′) system was reported by Yuan et al. [11]. It was constructed using the internally contracted multireference configuration interaction with a complete active space self-consistent field reference wavefunction and a neural network approach. The root mean square error of the fit is only 0.004 eV. The spectroscopic constants for the Chen PES are more accurate than those for the Prudente PES or Wernli PES [15]. In addition, the Chen PES is smooth over the whole configuration space, and there is no obvious barrier or well along the minimum-energy path (MEP), similar to the Wernli PES. Using the TDWP method, Yuan et al. carried out calculations for the ground rovibrational state of the reactant only. For the R1 channel, although the same QM method was used and no CSA was assumed, the integral crosssections (ICSs) obtained based on the Chen PES and the Wernli PES differ considerably for collision energies below 0.06 eV. The only reasonable explanation for this is the different features of the two PESs. Only the R1 reaction channel has been studied for the reactant in its ground rotational and vibrational states using the newest PES of Yuan et al. There are, as yet, no reliable results for the R2 channel. Furthermore, the most recent theoretical results on the effects of the rovibrational excitation of the reactant were obtained employing the Wernli PES. As the dynamical features of the Chen PES and the Wernli PES clearly differ in the low collision energy range and the CSA is not appropriate for this reaction, it is necessary to reinvestigate these features using this recently constructed PES (the Chen PES). This would also lead to a deeper understanding of the dynamics of the title reaction. Though it is common to use rigorous QM methods to treat triatomic problems [36–41], studies that use full quantum dynamics are numerically expensive [37,42,43], especially for calculations of the rovibrational excitations of reactants. However, the quasiclassical trajectory (QCT) method could be used instead to obtain some macrostatistical quantities [44]. Furthermore, the QCT method yields results that agree well with those obtained using QM methods for many systems [45], and it is useful for dissecting the various reaction mechanisms. Therefore, in the present study, the QCT method was employed to investigate the effects of rovibrational excitation of the reagent on the state-to-state reaction dynamics of the title reaction on the Chen PES. The paper is organized as follows. Computational details, the topography of the Chen PES, and an examination of the reliability of the QCT method for this reaction are presented in Section 2. The main results are discussed in detail in Section 3, and conclusions are drawn in Section 4.
Table 1 Values of bmax and ∊ for the two reaction channels as functions of Ec . Ec (eV)
0.022
0.043
0.130
0.217
0.304
0.390
0.477
0.564
bmax (Å) ∊ for R1 (% ) ∊ for R2 (% )
5.82 2.14 0.34
4.98 1.71 0.31
4.05 1.20 0.33
3.64 0.99 0.35
3.49 0.90 0.40
3.43 0.90 0.47
3.39 1.00 0.54
3.37 1.08 0.61
2.2. Topography of the PES In the entrance channel, the PES is attractive in all directions [11,12,19], and there is a well around the H atom of the reactant diatom. At larger separations of the reactant diatom (which may correspond to vibrational excitations of LiH), the well region spreads, so the PES is more attractive. In the exit channel, except for very small approach angles (defined as H′–H–Li) of less than 20°, the incoming H′ atom can always be abstracted by the H atom to form H2, because the PES slopes downhill along this channel. However, the topography of the PES along the R2 channel changes as the approach angle (defined as H′–Li–H) varies. When the approach angle is less than 140°, there is a shallow well in the interaction region. When the angle is larger than 140°, a small barrier appears. The shallow well may facilitate effective energy transfer in this channel. 2.3. Validity of the QCT method Because the R1 channel is dominant in the title reaction, and most of the calculations reported in the relevant literature were carried out for this channel, the TDWP results of Chen et al. [11] and our QCT results for this channel are compared in Fig. 1. The probabilities when b = 0 , the differential cross-sections (DCSs), the vibrationally resolved ICSs, and the total ICSs when b is sampled randomly are included. We can see that the QCT results agree fairly well with Chen’s results, except for the quantum resonance structure. The ICS results [11] obtained on the Wernli PES are also presented for comparison in Fig. 1d. The results from the Wernli PES in the lowenergy range (less than 0.04 eV) clearly deviate from those obtained with the Chen PES and QCT, as stated in Section 1. The excellent concordance between the TDWP and QCT results suggests that the QCT method can be used to calculate the reactivity in the energy range of this work for the title reaction.
2. Methods
3. Results and discussion
2.1. Computational details
In this section, we show the state-resolved features of the reactant and the product (the ICS, DCS, and state distribution of the product) for the two channels of the title reaction, and investigate the dynamical mechanism. In view of the fact that previous conclusions regarding this system have mainly been based on old PESs, and that the results obtained in different studies for the effects of the rovibrational excitation of the reactant contradict each other [18,19,29,26,13,23,22], these effects were reexamined using the new PES and the QCT method in the low-energy range.
Hamilton’s equations of motion were solved by the standard method of Karplus, Porter, and Sharma for atom-diatom collisions [46], which has been widely used to investigate the dynamics of chemical reactions [47–49]. Here, we only summarize the details pertinent to the present work. The QCT calculations were performed using the VENUS96 code [50,51]. For each collision energy (Ec ) and initial quantum state (v, j ) pair, the total number of trajectories Nt was 80000 and the time step for integration was set to 0.002 fs. The trajectories started on the PES at a distance of 10 Å between the incoming H′ and the center of mass (CM) of the LiH molecule, and terminated in the exit channel when R>10 Å. The maximum impact parameter, bmax , was simulated at each pair of Ec, v and j. The sampling error [52,53] was calculated according to
∊=
Nt − Nr , Nt Nr
3.1. Effect of rovibrational excitation of the reactant on the ICS 3.1.1. Effect of vibrational excitation ICSs obtained for various initial vibrational states (v = 0 − 4 , j = 0 ) in both channels (R1 and R2) are depicted in the first two panels of Fig. 2 using solid lines. For both channels, a strong dependence on vibrational excitation is apparent over the energy range considered. The reactivity of the R1 channel is greater than that of the R2 channel for lower vibrational excitations. For v = 4 , they are comparable. The ICS of the R1 channel drops more steeply as the vibrational quantum number v increases in the low-energy range. Vibrational excitation
(1)
where Nr is the number of trajectories for a specific reaction channel. The values of bmax and ∊ as functions of Ec with v = 0, j = 0 are listed in Table 1. 2
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Fig. 1. Comparisons of the total reaction probabilities, the DCSs, and the vibrationally resolved and total ICSs obtained using QM and QCT methods for the R1 channel. The solid lines are based on results from the Chen PES or Wernli PES, whereas lines with symbols derive from the present work.
Fig. 2. ICSs for various vibrational and rotational excitations in the two reaction channels. The solid lines are results obtained in the present work, the dotted and dashed lines are based on the results of Roy and Mahapatra [18], and the lines with symbols are from He et al. [19]. 3
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Fig. 3. DCSs for vibrational excitation at different collision energies.
due to the CSA used in the calculations performed by Roy et al. However, at collision energies lower than 0.06 eV, our results gradually begin to diverge from those of He et al. It seems that the differences between the PESs mainly affect the reaction dynamics in the low-energy range (refer to Fig. 6 in [11]). The only results that are available for the R2 channel are those of Roy et al. [18], in which the CSA is used. We plot those together with the present results for comparison here.
leads to a substantial increase in the ICS in the R2 channel. For the R1 channel, the collinear MEP is attractive and there is no barrier in either the reactant or the product channel. Vibrational excitation of LiH makes the PES more attractive on the Li side and generates more repulsive regions, meaning that more impinging H′ atoms rebound before reacting. This behavior makes the reaction in the R1 channel less likely to occur [29] and therefore reduces the reactivity [19]. In the R2 channel, for the v = 0 case, the ICS increases slowly up to approximately 0.3 eV and then gradually decreases. We speculate that this behavior occurs due to the shallow well in this channel, as stated in the analysis of the topography of the PES. The well permits effective (translational, vibrational, or rotational) energy transfer. This channel is thermoneutral. If the collision energy is relatively low, after some translational energy has been converted into rotational energy, the remaining translational energy is not enough to get the reactant past the well, so the ICS initially increases with increasing collision energy. However, when the collision energy is high enough, the direct mechanism causes the ICS to decrease with increasing collision energy. In contrast to the results for v = 0 at low collision energies, the ICSs for v = 1 − 4 show remarkably large values in the R2 channel. This phenomenon is termed ‘capture-type behavior’ with no threshold energy. It has also been found for other reactions [54,55], especially for the H + HBr system. The title reaction is similar to the H + HBr reaction in that the influence of the potential on the motion of the attacking H′ atom becomes more attractive at large separations of LiH. At the same time, because the gap between vibrational energy levels (about 0.17 eV) is of the same order of magnitude as the collision energy, vibrational excitation of LiH always helps H′ to pass through the well in this channel. In the first two panels of Fig. 2, the QM results of Roy et al. [18] (v = 0 − 3) and those of He et al. [19] (v = 0 − 1) obtained on the Wernli PES are shown as dotted or dashed lines and symbols, respectively. In the R1 channel, at collision energies higher than 0.06 eV, the effects of vibrational excitation are similar in three sets of results. The ICSs of He et al. [19] fit fairly well with the present QCT results but are slightly different in magnitude from those of Roy et al. [[18]], which is
3.1.2. Effect of rotational excitation Rotational excitation of the reactant does not have a significant influence on the dynamics of the title reaction according to our studies, as shown in the last two panels of Fig. 2, which agrees with the conclusions of recent literature [12,56]. This is due to the fact that the energy released from an excited rotational state is much lower. The rotational energy is not enough to have an obvious effect on the reactivity. That said, its effect can still be observed in our results; see, for example, the inset shown in Fig. 2. Generally speaking, rotational excitation decreases the reactivity of the R1 channel but increases the reactivity of the R2 channel. In the R1 channel, to form H2, the incoming H′ has to attack the H site. For this reaction system where the CM of LiH is near to the Li atom, rotational excitation of LiH causes the H atom to rapidly orbit Li, slightly obstructing it from reacting with the H′ atom. For the R2 channel, which is thermoneutral, rotational excitation helps to overcome the shallow well in the interaction region. In the last two panels of Fig. 2, ICSs for rotational excitation on the Wernli PES reported by Roy et al. [18] are presented as dotted or dashed lines. There are obvious differences between their and our results. In the R1 channel, if we only consider the trend, the magnitude of the ICS for j = 0 is about twice as large as the ICSs for the excited rotational states in Roy et al.’s study. In our results, however, those ICSs are almost undistinguishable. Furthermore, in the R2 channel, the trend in the ICS on the Wernli PES with increasing rotational excitation is the inverse of the trend seen in our results. The reason for this difference is unclear; it is difficult to determine whether it originates from the CSA or the PES itself.
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3.3. Effects of vibrational excitation on the rotational and vibrational state distributions of the product
3.2. Effect of vibrational excitation on the DCS The direction in which the product is scattered reveals more information about the reaction. Therefore, the distribution of the scattering angle of the product under vibrational excitation was calculated for both channels over the collision energy range considered here. The influence of rotational excitation on the DCS was found to be tiny, so only the results for vibrational excitation are presented here. In Fig. 3, the DCSs for vibrational quantum number v = 0 − 4 in both channels at three collision energies are depicted. At fixed energy, the magnitude of the DCS decreases in the R1 channel but increases in the R2 channel with increasing v, which is consistent with the trend observed for the ICSs in Fig. 2. For the R1 channel, a tendency for forward scattering is apparent with all v values, and this tendency strengthens with increasing collision energy. He et al. [19] arrived at the same conclusion based on the Wernli PES, but they only presented the results for v = 0, 1. Other literature that report the effects of vibrational excitation on the DCS are based on even older PESs, which will not be considered here. For the R2 channel, both forward and backward scattering are prominent, which is due to the shallow well in the interaction region. The tendency for backscattering dominates at higher collision energies. Furthermore, it is clear that the influence of vibrational excitation on the DCS curves decreases as the collision energy increases for both channels. This may be attributed to competition between the vibrational and collision energies. When the collision energy is lower than the vibrational energy, vibrational excitation plays a leading role in the reaction. When the collision energy exceeds the vibrational energy and takes the leading role in the reaction, the influence of vibrational excitation decreases.
Fig. 4 presents the channel-specific distributions of product rotational states obtained with the reactant in various vibrational states. We can see that in the R1 channel, the j′ value (rotational quantum number of the product) at the crest of the ICS curve is almost the same regardless of the value of v. The magnitude of this crest decreases markedly and the highest rotational quantum number that can be reached increases slightly with increasing v. This indicates that vibrational excitation of the reactant does not lead to increased rotational excitation of the product H2. The behavior of the ICS curve is, however, rather different in the R2 channel. As v increases, both the crest and the largest accessible j′ value increase. This indicates that vibrational excitation promotes the rotational excitation of the product LiH′. Similar conclusions can be drawn regarding the effect of vibrational excitation on the distribution of vibrational states of the product. Vibrational excitation of the reactant decreases the R1 channel but promotes the R2 channel, as presented in Fig. 5. According to our results, the influence of rotational excitation on the rotational and vibrational state distributions is tiny.
4. Conclusions Due to the different physical pictures obtained from the most extensively used PESs, especially at low collision energies, we used the QCT method to restudy the H + LiH reaction employing a newer PES. The validity of the QCT method has been tested in the R1 channel, and the excellent concordance between the TDWP and QCT results suggests that the QCT method can be used to calculated the reactivity in the energy range of this work. In this study, the influence of rovibrational excitation of the reactant
Fig. 4. Effects of vibrational excitation on the distribution of rotational states at different collision energies in the two reaction channels. 5
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Fig. 5. Effects of vibrational excitation on the distribution of vibrational states at different collision energies in the two reaction channels.
Declaration of Competing Interest
on the ICS, DCS, and the state distributions of the product was investigated. Both the LiH depletion and the H exchange channels were investigated in the low-energy range. Vibrational excitation of the reactant was found to have a significant effect, but the effect of rotational excitation was tiny. This is mainly because the gaps between vibrational energy levels are much larger than those between rotational energy levels. According to our ICS results, vibrational excitation reduces the reactivity of the R1 channel but promotes the reactivity of the R2 channel. The same behavior is observed with rotational excitation, although the influence of rotational excitation is not as strong as that of vibrational excitation. The dynamics below 0.06 eV are dramatically different from those obtained with the Wernli PES, and the effects of rotational excitation on the ICS that were observed in the present work differ significantly from those previously reported in the literature. Our DCS results indicate that forward scattering dominates for the R1 channel whereas backscattering is slightly favored for the R2 channel. The influence of vibrational excitation on the DCS is reduced at higher energies, which may be attributed to competition between the vibrational and collision energies. The products tend to be rotationally and vibrationally excited in both channels, and this tendency becomes more prominent at high collision energies. Vibrational excitation of the reactant reduced the rotational excitation and vibrational excitation of the product in the R1 channel but promotes those excitations in the R2 channel. In summary, we showed the dynamics of the reaction H + LiH in the low-energy range, which is dramatically different from those obtained with the old PESs. Further, the effects of rotational excitation based on the newer PES differ significantly from those previously reported in the literature.
None. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11374191, 11604333, and 21203108) and the National Basic Research Program of China (No. 2015CB921004). H. Yang is grateful for the support of the Taishan Scholars Project of Shandong Province (ts201712011). References [1] P.C. Stancil, A. Dalgarno, Stimulated radiative association of Li and H in the early universe, Astrophys. J. 479 (2) (1997) 543. [2] P.C. Stancil, S. Lepp, A. Dalgarno, The lithium chemistry of the early universe, Astrophys. J. 458 (1996) 401–406. [3] E. Bougleux, D. Galli, Lithium hydride in the early universe and in protogalactic clouds, Mon. Not. R. Astron. Soc. 288 (3) (1997) 638–648. [4] S. Lepp, P.C. Stancil, A. Dalgarno, Atomic and molecular processes in the early universe, J. Phys. B 35 (10) (2002) R57. [5] E. Bodo, F. Gianturco, R. Martinazzo, The gas-phase lithium chemistry in the early universe: elementary processes, interaction forces and quantum dynamics, Phys. Rep. 384 (3) (2003) 85–119. [6] R. Morales, M. Canales, A molecular hydrogen production model from Li and LiH in the early universe, Int. J. Astron. Astrophys. 3 (2) (2013) 108–112. [7] S. Gómez-Carrasco, L. González-Sánchez, N. Bulut, O. Roncero, L. Bañares, J.F. Castillo, State-to-state quantum wave packet dynamics of the LiH + H reaction on two ab initio potential energy surfaces, Astrophys. J. 784 (1) (2014) 55. [8] S. Bovino, M. Wernli, F.A. Gianturco, Fast LiH destruction in reaction with H: quantum calculations and astrophysical consequences, Astrophys. J. 699 (1) (2009) 383. [9] S. Bovino, M. Tacconi, F.A. Gianturco, D. Galli, F. Palla, On the relative abundance of LiH and LiH+ molecules in the early universe: new results from quantum reactions, Astrophys. J. 731 (2) (2011) 107. [10] M. Wernli, D. Caruso, E. Bodo, F.A. Gianturco, Computing a three-dimensional electronic energy manifold for the LiH + H →Li + H2 chemical reaction, J. Phys.
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Effects of rovibrational excitation of LiH on the LiH depletion and H exchange channels for the reaction H ( 2S) + LiH (X1Σ+) on a new potential energy surface Xiaolin Wanga,b, Yujun Zhenga, Huan Yanga, a b
T
⁎
School of Physics, Shandong University, Jinan 250100, China School of Physics and Optoelectronic Engineering, Weifang University, Weifang 261061, China
H I GH L IG H T S
of QCT Reproduces the Results of QM Very Well. • Method of Rotational and Vibrational Excitations to the Reaction Dynamics are Studied. • Effects of Dynamics are Shown for Energies Below 0.06 eV. • Results • Totally Different Effects of Rotational Excitations are Found.
A R T I C LE I N FO
A B S T R A C T
Keywords: QCT Vibrational excitation Rotational excitation PES
The quasiclassical trajectory method is used to calculate the dynamics of the reaction H + LiH for both reaction channels. The initial state-selected and energy-resolved integral cross-sections, differential cross-sections, and state distributions of the products are investigated in the collision energy range 0.02–0.56 eV. Results show that both vibrational and rotational excitations reduce the reactivity of the LiH depletion (R1) channel but promote that of the H exchange (R2) channel. Forward scattering is prominent in the R1 channel, but backscattering is slightly favored in the R2 channel. The influence of vibrational excitation is reduced at higher energies.
1. Introduction Due to the importance of the LiH molecule and the H′ + LiH reaction in early cosmochemistry [1–9], considerable effort has been directed into understanding the reaction dynamics of this reaction system from both quantum-mechanical (QM) [7,10–13,8,14–24] and quasiclassical [25–29,15,30–32] perspectives. The H + LiH reaction proceeds through the highly exothermic LiH depletion (R1) and thermoneutral H exchange (R2) channels,
H′ + LiH → Li + H2
(R1)
H′ + LiH → H+ LiH′
(R2)
A prerequisite for performing such studies is an accurate potential energy surface (PES) [33,34]. Since 2009, the most extensively used global three-dimensional PES for the title reaction has been the PES of Wernli et al. [10] (‘Wernli PES’ hereafter) and that of Prudente et al. [24] (‘Prudente PES’ hereafter). The main difference between these
⁎
PESs is that the Prudente PES has an early-stage energy barrier, whereas the Wernli PES does not [11]. The main advantage of the Wernli PES is said to be a more accurate description of long-range forces between the fragments and the elimination of the barrier in the entrance channel [7]. Because of these differences, the physical pictures obtained from these two PESs are obviously different, especially at low collision energies [11]. Various QM calculations have been carried out on the Wernli PES. Roy and Mahapatra studied the effect of rovibrational excitations of the reactant in their time-dependent wave packet (TDWP) calculations within the centrifugal sudden approximation (CSA) [18]. However, due to the barrierless nature of the title reaction, the CSA procedure is not appropriate for dynamical calculations in this system [19,35]. He et al. performed GPU-based calculations that accounted for the vibrational quantum number of the reactant (v = 0, 1) and treated the Coriolis coupling (CC) terms accurately [19], but only the R1 channel was taken into account. In 2015, a new global PES (‘Chen PES’ hereafter) for the lowest
Corresponding author. E-mail address: [email protected] (H. Yang).
https://doi.org/10.1016/j.cplett.2019.137043 Received 27 October 2019; Received in revised form 11 December 2019; Accepted 13 December 2019 Available online 28 December 2019 0009-2614/ © 2019 Published by Elsevier B.V.
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doublet state of the LiH2 ( 2A′) system was reported by Yuan et al. [11]. It was constructed using the internally contracted multireference configuration interaction with a complete active space self-consistent field reference wavefunction and a neural network approach. The root mean square error of the fit is only 0.004 eV. The spectroscopic constants for the Chen PES are more accurate than those for the Prudente PES or Wernli PES [15]. In addition, the Chen PES is smooth over the whole configuration space, and there is no obvious barrier or well along the minimum-energy path (MEP), similar to the Wernli PES. Using the TDWP method, Yuan et al. carried out calculations for the ground rovibrational state of the reactant only. For the R1 channel, although the same QM method was used and no CSA was assumed, the integral crosssections (ICSs) obtained based on the Chen PES and the Wernli PES differ considerably for collision energies below 0.06 eV. The only reasonable explanation for this is the different features of the two PESs. Only the R1 reaction channel has been studied for the reactant in its ground rotational and vibrational states using the newest PES of Yuan et al. There are, as yet, no reliable results for the R2 channel. Furthermore, the most recent theoretical results on the effects of the rovibrational excitation of the reactant were obtained employing the Wernli PES. As the dynamical features of the Chen PES and the Wernli PES clearly differ in the low collision energy range and the CSA is not appropriate for this reaction, it is necessary to reinvestigate these features using this recently constructed PES (the Chen PES). This would also lead to a deeper understanding of the dynamics of the title reaction. Though it is common to use rigorous QM methods to treat triatomic problems [36–41], studies that use full quantum dynamics are numerically expensive [37,42,43], especially for calculations of the rovibrational excitations of reactants. However, the quasiclassical trajectory (QCT) method could be used instead to obtain some macrostatistical quantities [44]. Furthermore, the QCT method yields results that agree well with those obtained using QM methods for many systems [45], and it is useful for dissecting the various reaction mechanisms. Therefore, in the present study, the QCT method was employed to investigate the effects of rovibrational excitation of the reagent on the state-to-state reaction dynamics of the title reaction on the Chen PES. The paper is organized as follows. Computational details, the topography of the Chen PES, and an examination of the reliability of the QCT method for this reaction are presented in Section 2. The main results are discussed in detail in Section 3, and conclusions are drawn in Section 4.
Table 1 Values of bmax and ∊ for the two reaction channels as functions of Ec . Ec (eV)
0.022
0.043
0.130
0.217
0.304
0.390
0.477
0.564
bmax (Å) ∊ for R1 (% ) ∊ for R2 (% )
5.82 2.14 0.34
4.98 1.71 0.31
4.05 1.20 0.33
3.64 0.99 0.35
3.49 0.90 0.40
3.43 0.90 0.47
3.39 1.00 0.54
3.37 1.08 0.61
2.2. Topography of the PES In the entrance channel, the PES is attractive in all directions [11,12,19], and there is a well around the H atom of the reactant diatom. At larger separations of the reactant diatom (which may correspond to vibrational excitations of LiH), the well region spreads, so the PES is more attractive. In the exit channel, except for very small approach angles (defined as H′–H–Li) of less than 20°, the incoming H′ atom can always be abstracted by the H atom to form H2, because the PES slopes downhill along this channel. However, the topography of the PES along the R2 channel changes as the approach angle (defined as H′–Li–H) varies. When the approach angle is less than 140°, there is a shallow well in the interaction region. When the angle is larger than 140°, a small barrier appears. The shallow well may facilitate effective energy transfer in this channel. 2.3. Validity of the QCT method Because the R1 channel is dominant in the title reaction, and most of the calculations reported in the relevant literature were carried out for this channel, the TDWP results of Chen et al. [11] and our QCT results for this channel are compared in Fig. 1. The probabilities when b = 0 , the differential cross-sections (DCSs), the vibrationally resolved ICSs, and the total ICSs when b is sampled randomly are included. We can see that the QCT results agree fairly well with Chen’s results, except for the quantum resonance structure. The ICS results [11] obtained on the Wernli PES are also presented for comparison in Fig. 1d. The results from the Wernli PES in the lowenergy range (less than 0.04 eV) clearly deviate from those obtained with the Chen PES and QCT, as stated in Section 1. The excellent concordance between the TDWP and QCT results suggests that the QCT method can be used to calculate the reactivity in the energy range of this work for the title reaction.
2. Methods
3. Results and discussion
2.1. Computational details
In this section, we show the state-resolved features of the reactant and the product (the ICS, DCS, and state distribution of the product) for the two channels of the title reaction, and investigate the dynamical mechanism. In view of the fact that previous conclusions regarding this system have mainly been based on old PESs, and that the results obtained in different studies for the effects of the rovibrational excitation of the reactant contradict each other [18,19,29,26,13,23,22], these effects were reexamined using the new PES and the QCT method in the low-energy range.
Hamilton’s equations of motion were solved by the standard method of Karplus, Porter, and Sharma for atom-diatom collisions [46], which has been widely used to investigate the dynamics of chemical reactions [47–49]. Here, we only summarize the details pertinent to the present work. The QCT calculations were performed using the VENUS96 code [50,51]. For each collision energy (Ec ) and initial quantum state (v, j ) pair, the total number of trajectories Nt was 80000 and the time step for integration was set to 0.002 fs. The trajectories started on the PES at a distance of 10 Å between the incoming H′ and the center of mass (CM) of the LiH molecule, and terminated in the exit channel when R>10 Å. The maximum impact parameter, bmax , was simulated at each pair of Ec, v and j. The sampling error [52,53] was calculated according to
∊=
Nt − Nr , Nt Nr
3.1. Effect of rovibrational excitation of the reactant on the ICS 3.1.1. Effect of vibrational excitation ICSs obtained for various initial vibrational states (v = 0 − 4 , j = 0 ) in both channels (R1 and R2) are depicted in the first two panels of Fig. 2 using solid lines. For both channels, a strong dependence on vibrational excitation is apparent over the energy range considered. The reactivity of the R1 channel is greater than that of the R2 channel for lower vibrational excitations. For v = 4 , they are comparable. The ICS of the R1 channel drops more steeply as the vibrational quantum number v increases in the low-energy range. Vibrational excitation
(1)
where Nr is the number of trajectories for a specific reaction channel. The values of bmax and ∊ as functions of Ec with v = 0, j = 0 are listed in Table 1. 2
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Fig. 1. Comparisons of the total reaction probabilities, the DCSs, and the vibrationally resolved and total ICSs obtained using QM and QCT methods for the R1 channel. The solid lines are based on results from the Chen PES or Wernli PES, whereas lines with symbols derive from the present work.
Fig. 2. ICSs for various vibrational and rotational excitations in the two reaction channels. The solid lines are results obtained in the present work, the dotted and dashed lines are based on the results of Roy and Mahapatra [18], and the lines with symbols are from He et al. [19]. 3
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Fig. 3. DCSs for vibrational excitation at different collision energies.
due to the CSA used in the calculations performed by Roy et al. However, at collision energies lower than 0.06 eV, our results gradually begin to diverge from those of He et al. It seems that the differences between the PESs mainly affect the reaction dynamics in the low-energy range (refer to Fig. 6 in [11]). The only results that are available for the R2 channel are those of Roy et al. [18], in which the CSA is used. We plot those together with the present results for comparison here.
leads to a substantial increase in the ICS in the R2 channel. For the R1 channel, the collinear MEP is attractive and there is no barrier in either the reactant or the product channel. Vibrational excitation of LiH makes the PES more attractive on the Li side and generates more repulsive regions, meaning that more impinging H′ atoms rebound before reacting. This behavior makes the reaction in the R1 channel less likely to occur [29] and therefore reduces the reactivity [19]. In the R2 channel, for the v = 0 case, the ICS increases slowly up to approximately 0.3 eV and then gradually decreases. We speculate that this behavior occurs due to the shallow well in this channel, as stated in the analysis of the topography of the PES. The well permits effective (translational, vibrational, or rotational) energy transfer. This channel is thermoneutral. If the collision energy is relatively low, after some translational energy has been converted into rotational energy, the remaining translational energy is not enough to get the reactant past the well, so the ICS initially increases with increasing collision energy. However, when the collision energy is high enough, the direct mechanism causes the ICS to decrease with increasing collision energy. In contrast to the results for v = 0 at low collision energies, the ICSs for v = 1 − 4 show remarkably large values in the R2 channel. This phenomenon is termed ‘capture-type behavior’ with no threshold energy. It has also been found for other reactions [54,55], especially for the H + HBr system. The title reaction is similar to the H + HBr reaction in that the influence of the potential on the motion of the attacking H′ atom becomes more attractive at large separations of LiH. At the same time, because the gap between vibrational energy levels (about 0.17 eV) is of the same order of magnitude as the collision energy, vibrational excitation of LiH always helps H′ to pass through the well in this channel. In the first two panels of Fig. 2, the QM results of Roy et al. [18] (v = 0 − 3) and those of He et al. [19] (v = 0 − 1) obtained on the Wernli PES are shown as dotted or dashed lines and symbols, respectively. In the R1 channel, at collision energies higher than 0.06 eV, the effects of vibrational excitation are similar in three sets of results. The ICSs of He et al. [19] fit fairly well with the present QCT results but are slightly different in magnitude from those of Roy et al. [[18]], which is
3.1.2. Effect of rotational excitation Rotational excitation of the reactant does not have a significant influence on the dynamics of the title reaction according to our studies, as shown in the last two panels of Fig. 2, which agrees with the conclusions of recent literature [12,56]. This is due to the fact that the energy released from an excited rotational state is much lower. The rotational energy is not enough to have an obvious effect on the reactivity. That said, its effect can still be observed in our results; see, for example, the inset shown in Fig. 2. Generally speaking, rotational excitation decreases the reactivity of the R1 channel but increases the reactivity of the R2 channel. In the R1 channel, to form H2, the incoming H′ has to attack the H site. For this reaction system where the CM of LiH is near to the Li atom, rotational excitation of LiH causes the H atom to rapidly orbit Li, slightly obstructing it from reacting with the H′ atom. For the R2 channel, which is thermoneutral, rotational excitation helps to overcome the shallow well in the interaction region. In the last two panels of Fig. 2, ICSs for rotational excitation on the Wernli PES reported by Roy et al. [18] are presented as dotted or dashed lines. There are obvious differences between their and our results. In the R1 channel, if we only consider the trend, the magnitude of the ICS for j = 0 is about twice as large as the ICSs for the excited rotational states in Roy et al.’s study. In our results, however, those ICSs are almost undistinguishable. Furthermore, in the R2 channel, the trend in the ICS on the Wernli PES with increasing rotational excitation is the inverse of the trend seen in our results. The reason for this difference is unclear; it is difficult to determine whether it originates from the CSA or the PES itself.
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3.3. Effects of vibrational excitation on the rotational and vibrational state distributions of the product
3.2. Effect of vibrational excitation on the DCS The direction in which the product is scattered reveals more information about the reaction. Therefore, the distribution of the scattering angle of the product under vibrational excitation was calculated for both channels over the collision energy range considered here. The influence of rotational excitation on the DCS was found to be tiny, so only the results for vibrational excitation are presented here. In Fig. 3, the DCSs for vibrational quantum number v = 0 − 4 in both channels at three collision energies are depicted. At fixed energy, the magnitude of the DCS decreases in the R1 channel but increases in the R2 channel with increasing v, which is consistent with the trend observed for the ICSs in Fig. 2. For the R1 channel, a tendency for forward scattering is apparent with all v values, and this tendency strengthens with increasing collision energy. He et al. [19] arrived at the same conclusion based on the Wernli PES, but they only presented the results for v = 0, 1. Other literature that report the effects of vibrational excitation on the DCS are based on even older PESs, which will not be considered here. For the R2 channel, both forward and backward scattering are prominent, which is due to the shallow well in the interaction region. The tendency for backscattering dominates at higher collision energies. Furthermore, it is clear that the influence of vibrational excitation on the DCS curves decreases as the collision energy increases for both channels. This may be attributed to competition between the vibrational and collision energies. When the collision energy is lower than the vibrational energy, vibrational excitation plays a leading role in the reaction. When the collision energy exceeds the vibrational energy and takes the leading role in the reaction, the influence of vibrational excitation decreases.
Fig. 4 presents the channel-specific distributions of product rotational states obtained with the reactant in various vibrational states. We can see that in the R1 channel, the j′ value (rotational quantum number of the product) at the crest of the ICS curve is almost the same regardless of the value of v. The magnitude of this crest decreases markedly and the highest rotational quantum number that can be reached increases slightly with increasing v. This indicates that vibrational excitation of the reactant does not lead to increased rotational excitation of the product H2. The behavior of the ICS curve is, however, rather different in the R2 channel. As v increases, both the crest and the largest accessible j′ value increase. This indicates that vibrational excitation promotes the rotational excitation of the product LiH′. Similar conclusions can be drawn regarding the effect of vibrational excitation on the distribution of vibrational states of the product. Vibrational excitation of the reactant decreases the R1 channel but promotes the R2 channel, as presented in Fig. 5. According to our results, the influence of rotational excitation on the rotational and vibrational state distributions is tiny.
4. Conclusions Due to the different physical pictures obtained from the most extensively used PESs, especially at low collision energies, we used the QCT method to restudy the H + LiH reaction employing a newer PES. The validity of the QCT method has been tested in the R1 channel, and the excellent concordance between the TDWP and QCT results suggests that the QCT method can be used to calculated the reactivity in the energy range of this work. In this study, the influence of rovibrational excitation of the reactant
Fig. 4. Effects of vibrational excitation on the distribution of rotational states at different collision energies in the two reaction channels. 5
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Fig. 5. Effects of vibrational excitation on the distribution of vibrational states at different collision energies in the two reaction channels.
Declaration of Competing Interest
on the ICS, DCS, and the state distributions of the product was investigated. Both the LiH depletion and the H exchange channels were investigated in the low-energy range. Vibrational excitation of the reactant was found to have a significant effect, but the effect of rotational excitation was tiny. This is mainly because the gaps between vibrational energy levels are much larger than those between rotational energy levels. According to our ICS results, vibrational excitation reduces the reactivity of the R1 channel but promotes the reactivity of the R2 channel. The same behavior is observed with rotational excitation, although the influence of rotational excitation is not as strong as that of vibrational excitation. The dynamics below 0.06 eV are dramatically different from those obtained with the Wernli PES, and the effects of rotational excitation on the ICS that were observed in the present work differ significantly from those previously reported in the literature. Our DCS results indicate that forward scattering dominates for the R1 channel whereas backscattering is slightly favored for the R2 channel. The influence of vibrational excitation on the DCS is reduced at higher energies, which may be attributed to competition between the vibrational and collision energies. The products tend to be rotationally and vibrationally excited in both channels, and this tendency becomes more prominent at high collision energies. Vibrational excitation of the reactant reduced the rotational excitation and vibrational excitation of the product in the R1 channel but promotes those excitations in the R2 channel. In summary, we showed the dynamics of the reaction H + LiH in the low-energy range, which is dramatically different from those obtained with the old PESs. Further, the effects of rotational excitation based on the newer PES differ significantly from those previously reported in the literature.
None. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11374191, 11604333, and 21203108) and the National Basic Research Program of China (No. 2015CB921004). H. Yang is grateful for the support of the Taishan Scholars Project of Shandong Province (ts201712011). References [1] P.C. Stancil, A. Dalgarno, Stimulated radiative association of Li and H in the early universe, Astrophys. J. 479 (2) (1997) 543. [2] P.C. Stancil, S. Lepp, A. Dalgarno, The lithium chemistry of the early universe, Astrophys. J. 458 (1996) 401–406. [3] E. Bougleux, D. Galli, Lithium hydride in the early universe and in protogalactic clouds, Mon. Not. R. Astron. Soc. 288 (3) (1997) 638–648. [4] S. Lepp, P.C. Stancil, A. Dalgarno, Atomic and molecular processes in the early universe, J. Phys. B 35 (10) (2002) R57. [5] E. Bodo, F. Gianturco, R. Martinazzo, The gas-phase lithium chemistry in the early universe: elementary processes, interaction forces and quantum dynamics, Phys. Rep. 384 (3) (2003) 85–119. [6] R. Morales, M. Canales, A molecular hydrogen production model from Li and LiH in the early universe, Int. J. Astron. Astrophys. 3 (2) (2013) 108–112. [7] S. Gómez-Carrasco, L. González-Sánchez, N. Bulut, O. Roncero, L. Bañares, J.F. Castillo, State-to-state quantum wave packet dynamics of the LiH + H reaction on two ab initio potential energy surfaces, Astrophys. J. 784 (1) (2014) 55. [8] S. Bovino, M. Wernli, F.A. Gianturco, Fast LiH destruction in reaction with H: quantum calculations and astrophysical consequences, Astrophys. J. 699 (1) (2009) 383. [9] S. Bovino, M. Tacconi, F.A. Gianturco, D. Galli, F. Palla, On the relative abundance of LiH and LiH+ molecules in the early universe: new results from quantum reactions, Astrophys. J. 731 (2) (2011) 107. [10] M. Wernli, D. Caruso, E. Bodo, F.A. Gianturco, Computing a three-dimensional electronic energy manifold for the LiH + H →Li + H2 chemical reaction, J. Phys.
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