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The effects of collision energy and reagent vibrational excitation on the reactivity of the reaction H + LiH: A quasiclassical trajectory study
Computational and Theoretical Chemistry 1006 (2013) 31–36
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Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
The effects of collision energy and reagent vibrational excitation on the reactivity of the reaction H + LiH: A quasiclassical trajectory study Zhi-Jun Jiang, Mei-Shan Wang ⇑, Chuan-Lu Yang, Di He School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, People’s Republic of China
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Article history: Received 12 July 2012 Received in revised form 3 December 2012 Accepted 6 December 2012 Available online 27 December 2012 Keywords: Reagent vibrational excitation Collision energy QCT Reactivity
a b s t r a c t Quasi-classical trajectory (QCT) calculations are carried out for the reaction H + LiH (m = 0–3, j = 0) using a recent analytical potential energy surface (PES) of Prudente et al. [F.V. Prudente, J.M.C. Marques, A.M. Maniero, Time-dependent wave packet calculation of the LiH + H reactive scattering on a new potential energy surface, Chem. Phys. Lett. 474 (2009) 18–22]. The reaction H + LiH proceeds through the highly exothermic LiH depletion and thermoneutral hydrogen-exchange channels. State-selected and energyresolved reaction probability and integral reaction cross sections are reported and compared with the recent available literature data. The QCT-calculated probabilities are in good agreement with the previous quantum–mechanical (QM) results, meanwhile, the vibrational excitation of the reagent LiH molecule decreases the reactivity of the LiH depletion channel and increases the reactivity of the hydrogenexchange channel. Moreover, this decrease and increase can be explained by examining the representative reactive trajectories for the different vibrational excitation of the reagent and the topology of the HHLi PES. Our calculated results indicate that the collision energy and the vibrational excitation of the reagent play an important role on the reactivity of this system. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The lithium chemistry has recently received considerable attention due to the important role of LiH molecule and its ionic variants in the primordial universe [1–6]. Lithium chemistry begins with the radiative association reactions, which form LiH and LiH+ species. According to the standard Big Bang nucleosynthesis model, the early universe was very simple, it only had electrons, low energy photons and the lightest nuclei H, D, He and Li. The LiH molecule and its ionic variants took part in the chemical reaction network in an entirely gaseous phase [7,8]. In particular, the formation and depletion of LiH and LiH+ species are thought to be able to leave the imprint of their existence on the spectrum of the cosmic background radiation (CBR) from remote epochs [9]. LiH is known to be formed through radiative association and depleted by reactions with atomic H to form H2. Up to now, the experimental investigation of the LiH formation reaction in the different electronic states have been reported [10,11]. Chen et al. [10] observed the nascent rotational population distribution of LiH (m = 0) in the reaction Li (22 PJ) + H2. Bililign et al. [11] studied the product LiH of the reaction Li + H2 through the laser pump–probe far-wing scattering experiment. Theoretical studies on LiH molecule including the spectra, polarizability and mono-bichromatic electron dynamics also have been reported [12–14]. ⇑ Corresponding author. Tel./fax: +86 535 6672142. E-mail address: [email protected] (M.-S. Wang). 2210-271X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2012.12.006
Despite the low number of electrons, the LiH2 system shows all the complex features of the more complicated chemical species. In recent decades, considerable effort has been given to constructing the PES of LiH2 system, investigating the sub-reactive [17] and reactive [22–27] collision dynamics which can be carried out using the constructed PES. An accurate electronic-state PES is necessary in order to examine its reactive dynamics. Several PESs of the electronic ground state of LiH2 system have been developed by the leading experts in this field [15–21]. The PES of the ground state of LiH2 system was constructed by Clarke et al. [15] for the collinear arrangement through the application of the spin-coupled valence bond non-orthogonal configuration interaction (SCVBNOCI) method. Their results showed that there existed an early small barrier of about 0.036 eV in the LiH + H entrance channel. Subsequent to Clarke et al.’s work, two PESs of LiH2 were reported in literature [16,17]. The first three-dimensional PES was developed by Dunne, Murrell and Jemmer (DMJ) [16] by fitting the configuration interaction (CI) ab initio points to an analytical function within the manybody expansion (MBE) framework. Contrary to what was reported by Clarke et al. [15], this PES did not show any barriers in the LiH + H entrance channel. The other one was developed by Bodo et al. [17] through applying the coupled-cluster method with single, double and triple excitations (CCSD(T)) to study the nonreactive dynamics. In order to study the properties of reactive dynamics, a more complete theoretical PES for the ground state was obtained by Kim et al. [18], using an expanded basis set and the sophisticated interpolation method. Berriche and Tlili [19]
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presented a study of the interaction between hydrogen atom and LiH molecule. In 2009, Wernli et al. [20] constructed a threedimensional PES for the electronic ground state of the reaction H + LiH ? H2 + Li by using the complete active space self consistent field (CASSCF) method followed by the multireference configuration interaction (MRCI) treatment of the correlation energy. The exothermicity of the reaction was estimated to be 2.258 eV [20], consistent with the value of 2.23 eV obtained by Dunne et al. [16]. The barrier to the reaction was also found to be absent on the PES of Wernli et al. [20]. The van der Waals well was found in the product valley of Li + H2 on the DMJ PES [16], but it was found to be absent on the PES of Wernli et al. [20]. Later that same year, Prudente et al. [21] also developed an accurate PES of the ground electronic state for the reaction H + LiH and performed the quantum wave packet calculations on it. Their calculations indicated that the Prudente PES clearly improved the DMJ PES [16] in several ways and was much smoother [21]. The dynamics of H + LiH collisions was also studied on the constructed PESs by the QM and QCT calculations [22–29]. In the recent past, a series of QM calculations on the DMJ PES [16] have been carried out to investigate the dynamics of this reaction in detail [22–27]. Defazio et al. [22] and Padmanaban and Mahapatra [23–27] applied the QM method to study the reaction probability, reaction cross section, thermal rate constant, mechanism and resonance of H + LiH scattering. Liu et al. [28] calculated the stereodynamics of the reaction H + LiH (v = 0, j = 0) ? H2 + Li by using QCT method based on the PES of Prudente et al. [21]. Moreover, Roy and Mahapatra [29] applied the time-dependent quantum wave packet dynamics method to study the initial state-selected channelspecific reactivity of H + LiH collisional system and its isotopic variants on the ab initio PES developed by Wernli et al. [20]. It was also worthy to note that the ionic variants of LiH2 system is the same important as the neutral system and shares many common features with the neutral reaction. The recent work on this subject was the quantum dynamics study of H + LiH+ reaction reported by Roy et al. [30]. They reported the channel-specific integral reaction cross-sections and thermal rate constants of the reaction H + LiH+. As mentioned above, most of the previous experimental and theoretical studies for the title reaction were focused on the rotational population distribution of LiH, reaction probability, cross section, rate constant, and resonance of H + LiH (v = 0, j = 0) collisions. However, there are few studies [25,29] related to the effects of the vibrational and rotational excitation of the reagent on the reactivity of H + LiH reaction. In this paper, we apply the QCT method to study the effects of the collision energy and vibrational excitation of the reagent on the reactivity of the title reaction on the Prudente PES [21]. We focus here on the dynamics of H + LiH reaction which can proceed via the following two reactive paths:
a standard Monte Carlo sampling of the initial conditions and the adaptive-step Adams method is used to integrate Hamilton’s equations. The conservations of both the total energy and total angular momentum have been checked carefully. In the calculation, the classical Hamilton’s equations are numerically integrated for motion in three dimensions, and the accuracy of the integration is verified by checking the conservation of the total energy and total angular momentum for every trajectory. In the present work, the vibrational and rotational levels of the reactant molecule are taken as m = 0–3, j = 0, respectively. Reaction probabilities and cross sections as a function of collision energy for the H + LiH reaction have been calculated by running a batch of 50,000 trajectories at the randomly and uniformly sampled collision energies within the range 0.001–1.0 eV. The trajectories are started at an initial distance of 15 Å between H atom and the center of the mass of LiH molecule. The total reaction probability Nr/Nt for total angular momentum J = 0 (the impact parameter equals to 0) is the ratio of the number of reactive trajectories Nr to the total number of trajectories Nt. In the QCT framework, for a fixed collision energy Ec and the given rovibrational state (v, j) of LiH, the reaction cross section rvj can be written as:
rv j ¼ pb2max ðNr =Nt Þ;
ð1Þ
where bmax denotes the maximum value of the reactive impact parameter for the collision energy Ec. 3. Results and discussion The QCT reaction probabilities for the LiH depletion (H + LiH (m = 0, j = 0) ? H2 + Li) and H-exchange (Ha + LiHb (m = 0, j = 0) ? LiHa + Hb) channels with J = 0 are displayed in Fig. 1. In order to certify the validity of the QCT method, we firstly compared the QCT probabilities with the previous quantum results, which can be also found in Fig. 1. The red line and blue line are quantum results reported by Prudente et al. [21] and Roy and Mahapatra [29], respectively. Prudente et al. [21] only calculated the reaction probabilities for collision energies between 0.1 and 1.0 eV. It is clear from Fig. 1 that the QCT results are in general agreement with the quantum results. The PES used by us is same with the one used by Prudente. The QCT results are slightly smaller than the QM results, it may be due to the lack of the quantum effects in QCT method. After certifying the effectiveness of the QCT method for this system, we investigate the effect of the reagent vibrational excitation
H þ LiH ! H2 þ Li ðR1Þ; Ha þ LiHb ! LiHa þ Hb ðR2Þ: Although the highly exothermic LiH depletion path (R1) is considered to contribute to LiH depletion, the thermoneutral hydrogen exchange path (R2) is thought to lead to the retention of LiH [23]. Besides, there is another non-reactive path leading to the retention of LiH. The paper is organized as follows: in Section 2 we briefly review the basic theory. In Section 3, results and discussion are presented. Finally, Section 4 gathers the conclusions. 2. Basic theory Details of the QCT method can be found in Refs. [31–34] and only a brief description is given here. The QCT calculations employ
Fig. 1. Comparison of the calculated H + LiH (m = 0, j = 0) reaction probabilities for total angular momentum J = 0 between the QCT (this work) and QM (cf. Fig. 3 of Ref. [21] and Fig. 1 of Ref. [29]) calculations of LiH depletion (R1) and hydrogen exchange (R2) channels.
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Fig. 2. QCT total reaction probabilities of the H + LiH (m, j = 0) reaction as a function of collision energy for various vibrational excitation states of the R1 and R2 channels.
on the reaction probabilities of R1 and R2 channels. The J = 0 total reaction probabilities for the various vibrational quantum state of LiH molecule (m = 0–3) are plotted as a function of collision energies and the results are presented in Fig. 2. It can be seen from Fig. 2 that the vibrational excitation of the reagent LiH causes a substantial reduction of the probabilities of R1 channel, whereas it causes a substantial increase of the probabilities of the R2 channel. The extent of decrease and increase of the two channels are quite different between v = 0 and v – 0. In addition, it is observed that the exothermic LiH depletion reaction is more efficient than the thermoneutral H-exchange reaction. Similar findings were also reported by Wernli et al. [20], Prudente et al. [21] and Roy and Mahapatra [29]. The QCT integral cross sections of the reaction H + LiH (m, j = 0) for the R1 and R2 channels are presented in Figs. 3a and 4a. The reaction cross sections for the various vibrational quantum state of LiH molecule (m = 0–3) are plotted as a function of collision energies. In the given energy range, Roy and Mahapatra [29] accurately calculated the partial wave contribution for total angular momentum up to J = 70. They found that J = 42 is necessary to obtain the
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converged cross sections for the R1 and R2 reactive channels, respectively. We find that the reaction cannot be carried out below the collision energy of 0.001 eV and begins to happen at about 0.002 eV. It means that the PES of LiH2 system exists a very small barrier, which causes a nonzero threshold in the reaction H + LiH. It is important to note that Liu et al. [28] found a small barrier at the collinear arrangements of LiH2 system. The PES used by us is same with the one used by Liu et al. A very small barrier appears along the path for the LiH depletion reaction, which is basically consistent with the barrier-less description in Ref. [15]. It can be seen from Fig. 3a that the QCT cross sections decrease with the increase of the collision energy, as expected for a barrier-less highly exothermic reaction. Moreover, it is clear from Fig. 3a that the vibrational excitation of the reagent LiH leads to a substantial reduction of its depletion cross section, whereas it results in a substantial increase of its H-exchange reactivity [cf. Fig. 4a]. Namely, the vibrational excitation of the reagent LiH has a negative effect on the reactivity of the LiH depletion channel, whereas it promotes the reactivity of the H-exchange channel. These trends are consistent with the earlier findings [25,29] (cf. Fig. 7 of Ref. [25] and Fig. 4 of Ref. [29]) on the PESs from DMJ and Wernli et al. Polanyi and coworkers [35] have extensively studied the effects of the reagent vibrational and rotational excitation on the reactivity of a chemical reaction system. Generally, the vibrational excitation of the reagent can promote the reactivity of R1 channel while the PES exists a late barrier [35]. The minimum energy paths (MEPs) of the collinear geometry for R1 channel of the title reaction is attractive [28] and no barrier is found in the exit valley [21]. Therefore, it is reasonable that the vibrational excitation of the reagent inhibits the reactivity of the depletion channel. The dynamics of the R2 reactive channel seems to be more complicated. It has been showed that the R2 channel exists the potential basin along the collinear minimum energy path, which appears at the distance of the closest approach of the reagents and is much deeper on the DMJ PES [16]. The potential basin permits an effective transfer of the translational and vibrational energies to the R2 reactive channel [36], therefore, the reagent approaches each other and further to bind together at the distance of the closest approach. Similar findings were also reported by Roy and Mahapatra [29]. From Figs. 3a and 4a, we can also conclude that the effects of the reagent vibrational excitation on the cross sections seems to be more marked at the
Fig. 3. The reaction cross section of the reaction H + LiH (m = 0–3, j = 0) ? H2 + Li as a function of collision energy for the QCT results from this work are compared with QM results of Ref. [29].
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Fig. 4. The reaction cross section of the reaction Ha + LiHb (m = 0–3, j = 0) ? LiHa + Hb as a function of collision energy for the QCT results from this work are compared with QM results of Ref. [29].
lower collision energy than at the higher collision energy. The QM results of Roy and Mahapatra [29] are also shown for comparison in Figs. 3b and 4b. When the values of QCT depletion cross sections are expanded 1.5 times, the shape of the present QCT reaction cross sections is very similar to that of Roy et al. although the magnitude
Fig. 5. Representative reactive trajectories for R1 for LiH vibration states
of the reaction cross sections is somewhat smaller than the Roy’s result. These differences are probably as a result of the different PESs and the lack of the quantum tunneling in the QCT method. But overall, the trend of the present QCT results are in good agreement with the available QM results.
v = 0–3, as plotted on the collinear HHLi surface at the collision energy of 0.05 eV and ht = 43°.
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Fig. 6. Representative reactive trajectories for R1 for LiH vibration states
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v = 0–3, as plotted on the collinear HHLi surface at the collision energy of 0.2 eV and ht = 69°.
The reactivity of the reaction H + LiH can be explained by examining the representative reactive trajectories for the different vibrational excitation, and the topology of the HHLi PES as shown in Figs. 5 and 6. To obtain a clearer understanding of the reaction mechanism, we observe lots of representative reactive trajectories which plot on the collinear HHLi surface. The contours of PES are constructed as the function of RHH and RLiH. It is worth noting that we find a very small barrier for the collinear path and it is perfectly consistent with the observations of Liu et al. [28]. The trajectories of the depletion channel with the variation of internuclear distances Ha–Hb and Hb–Li for the H + LiH (m, j = 0) reaction, which plot on the collinear HHLi surface, are also presented in Figs. 5 and 6. The trajectories are selected in scattering angle ranges of ht = 43° and 69° at the lower collision energy of 0.05 and 0.2 eV. In Fig. 5a, the initial state selected trajectories of the reaction H + LiH (m = 0, j = 0) show an interesting phenomenon at the collision energy of 0.05 eV. We observe many indirect reactive trajectories, which are called ‘‘trapped’’ trajectories. Although the PESs of DMJ [16] and Prudente et al. [21] are both highly attractive along the collinear minimum energy path of the LiH depletion channel, the DMJ PES supports two unphysical wells, which appear in the reagent and product channels along the minimum energy path on the DMJ PES [16,29]. Owing to the Prudente PES [21] was achieved by performing accurate ab initio full configuration interaction calculations and fitting them to a slightly modified version of the analytical functions of DMJ [16], the existence of unphysical wells in the entrance or exit valleys make the longlived trajectories emerge on both PESs. It is clear from Fig. 5a that the projectile atom H undergoes several collisions with the target molecule LiH and the complexes have relatively long lifetimes
before breaking-up into one of the random channels. The relatively long-lived survival of the metastable states could permit the system to perform the extensive rotations before breaking up, therefore make the products appear in the more random directions. Fig. 5b and c present a non-reactive trajectories, which implies that the vibrational excitation of the reagent decreases the reactivity of the R1 channel. The results reasonably explain why the vibrational excitation of the reagent causes a substantial reduction of cross section of this system. To our surprise, in Fig. 5d we observe the H-exchange channel trajectory of the reaction H + LiH (for initial m = 3, j = 0) at the collision energy of 0.05 eV. In other words, the vibrational excitation of the reagent LiH causes a substantial reduction of its depletion reactivity, whereas it causes a substantial increase of its exchange reactivity. Fig. 6a and b show that the H atom collides with the LiH molecule and forms a H2 molecule that moves away immediately at the collision energy of 0.2 eV and ht = 69°. This process is in conformity with the dynamics of the direct reaction. Besides, at the higher collision energy range, we do not find any trajectories of the indirect reaction. We only show the representative trajectories at Ec = 0.05 and 0.2 eV here. In most cases, the product molecule H2 moves away immediately when the attacking atom H collides with the LiH molecule, and the collision process is relatively short-lived. Overall, the H + LiH collisions is mainly dominated by the direct reaction mechanism, but the indirect mechanism plays a role while the collision energy is lower. As can be seen in Fig. 6c and d, the attacking atom Ha collides with the LiHb molecule and forms a LiHa molecule that moves away immediately. That is to say, the vibrational excitation of the reagent plays an important role on the reactivity of the reaction H + LiH. It also illustrates that the vibrational excitation of the reagent
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LiH inhibits the reactivity of the depletion channel, whereas it obviously promotes the reactivity of the H-exchange channel. The corresponding QCT results for the initial state m = 0–3 and j = 0 are in good agreement with the available QM results from the recent literature [29]. 4. Conclusion In this paper, QCT calculations have been performed to study the effects of the collision energy and the vibrational excitation of the reagent on the reactivity of H + LiH reaction. We first calculate the J = 0 total reaction probabilities of both the LiH depletion channel and H-exchange channel, and the QCT results are compared with the recent available literature data. The reactivity of the LiH depletion channel is found to be greater than the H-exchange channel, which is very similar to the reaction H + LiH+ reported by Roy et al. [30]. Meanwhile, the present QCT results are in good accord with the available QM results from the recent literature. On this basis, we investigate the influence of the collision energy and the vibrational excitation of the reagent on the reaction cross sections of the H + LiH reaction. The reaction cross sections decrease with increasing collision energy. While we find that if v – 0 and j = 0, the vibrational excitation of the reagent LiH decreases the reactivity of the depletion channel and increases the reactivity of the H-exchange channel. Such a trend has been also pointed out in a recent work [29]. This phenomenon can be explained by examining the representative reactive trajectories for the different vibrational excitation on the collinear HHLi surface. The calculated results imply that the collision energy and the vibrational excitation of the reagent play an important role on the reactivity of this system. It is important to note that the above findings also have further confirmed the reliability of the results obtained by Roy and Mahapatra [29]. Acknowledgements The authors are very grateful to Professor Frederico V. Prudente for providing the codes of the potential energy surface [21]. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11074103, 10974078, and 11174117) and Discipline Construction Fund of Ludong University. All calculations were carried out in the Shuguang Super Computer Center (SSCC) of Ludong University. The authors also appreciate Professor Han for providing the QCT code. References [1] S. Lepp, J. Shull, Molecules in the early universe, Astrophys. J. 280 (1984) 465– 469. [2] N. Yoshida, T. Abel, L. Hernquist, N.S. Yoshida, Simulations of early structure formation: primordial gas clouds, Astrophys. J. 592 (2003) 645–663. [3] T. Prodanovic´, B.D. Fields, On nonprimordial deuterium production by accelerated particles, Astrophys. J. 597 (2003) 48–56. [4] T. Prodanovic, B.D. Fields, Probing primordial and pre-galactic lithium with high-velocity clouds, Astrophys. J. 616 (2004) L115–L118. [5] S. Lepp, P.C. Stancil, A. Dalgarno, Atomic and molecular processes in the early Universe, J. Phys. B: At., Mol. Opt. Phys. 35 (2002) R57–R80. [6] E. Bodo, F.A. Gianturco, R. Martinazzo, The gas-phase lithium chemistry in the early universe: elementary processes, interaction forces and quantum dynamics, Phys. Rep. 384 (2003) 85–119. [7] D. Galli, F. Palla, The chemistry of the early Universe, Astron. Astrophys. 335 (1998) 403–420.
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Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
The effects of collision energy and reagent vibrational excitation on the reactivity of the reaction H + LiH: A quasiclassical trajectory study Zhi-Jun Jiang, Mei-Shan Wang ⇑, Chuan-Lu Yang, Di He School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 12 July 2012 Received in revised form 3 December 2012 Accepted 6 December 2012 Available online 27 December 2012 Keywords: Reagent vibrational excitation Collision energy QCT Reactivity
a b s t r a c t Quasi-classical trajectory (QCT) calculations are carried out for the reaction H + LiH (m = 0–3, j = 0) using a recent analytical potential energy surface (PES) of Prudente et al. [F.V. Prudente, J.M.C. Marques, A.M. Maniero, Time-dependent wave packet calculation of the LiH + H reactive scattering on a new potential energy surface, Chem. Phys. Lett. 474 (2009) 18–22]. The reaction H + LiH proceeds through the highly exothermic LiH depletion and thermoneutral hydrogen-exchange channels. State-selected and energyresolved reaction probability and integral reaction cross sections are reported and compared with the recent available literature data. The QCT-calculated probabilities are in good agreement with the previous quantum–mechanical (QM) results, meanwhile, the vibrational excitation of the reagent LiH molecule decreases the reactivity of the LiH depletion channel and increases the reactivity of the hydrogenexchange channel. Moreover, this decrease and increase can be explained by examining the representative reactive trajectories for the different vibrational excitation of the reagent and the topology of the HHLi PES. Our calculated results indicate that the collision energy and the vibrational excitation of the reagent play an important role on the reactivity of this system. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The lithium chemistry has recently received considerable attention due to the important role of LiH molecule and its ionic variants in the primordial universe [1–6]. Lithium chemistry begins with the radiative association reactions, which form LiH and LiH+ species. According to the standard Big Bang nucleosynthesis model, the early universe was very simple, it only had electrons, low energy photons and the lightest nuclei H, D, He and Li. The LiH molecule and its ionic variants took part in the chemical reaction network in an entirely gaseous phase [7,8]. In particular, the formation and depletion of LiH and LiH+ species are thought to be able to leave the imprint of their existence on the spectrum of the cosmic background radiation (CBR) from remote epochs [9]. LiH is known to be formed through radiative association and depleted by reactions with atomic H to form H2. Up to now, the experimental investigation of the LiH formation reaction in the different electronic states have been reported [10,11]. Chen et al. [10] observed the nascent rotational population distribution of LiH (m = 0) in the reaction Li (22 PJ) + H2. Bililign et al. [11] studied the product LiH of the reaction Li + H2 through the laser pump–probe far-wing scattering experiment. Theoretical studies on LiH molecule including the spectra, polarizability and mono-bichromatic electron dynamics also have been reported [12–14]. ⇑ Corresponding author. Tel./fax: +86 535 6672142. E-mail address: [email protected] (M.-S. Wang). 2210-271X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2012.12.006
Despite the low number of electrons, the LiH2 system shows all the complex features of the more complicated chemical species. In recent decades, considerable effort has been given to constructing the PES of LiH2 system, investigating the sub-reactive [17] and reactive [22–27] collision dynamics which can be carried out using the constructed PES. An accurate electronic-state PES is necessary in order to examine its reactive dynamics. Several PESs of the electronic ground state of LiH2 system have been developed by the leading experts in this field [15–21]. The PES of the ground state of LiH2 system was constructed by Clarke et al. [15] for the collinear arrangement through the application of the spin-coupled valence bond non-orthogonal configuration interaction (SCVBNOCI) method. Their results showed that there existed an early small barrier of about 0.036 eV in the LiH + H entrance channel. Subsequent to Clarke et al.’s work, two PESs of LiH2 were reported in literature [16,17]. The first three-dimensional PES was developed by Dunne, Murrell and Jemmer (DMJ) [16] by fitting the configuration interaction (CI) ab initio points to an analytical function within the manybody expansion (MBE) framework. Contrary to what was reported by Clarke et al. [15], this PES did not show any barriers in the LiH + H entrance channel. The other one was developed by Bodo et al. [17] through applying the coupled-cluster method with single, double and triple excitations (CCSD(T)) to study the nonreactive dynamics. In order to study the properties of reactive dynamics, a more complete theoretical PES for the ground state was obtained by Kim et al. [18], using an expanded basis set and the sophisticated interpolation method. Berriche and Tlili [19]
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presented a study of the interaction between hydrogen atom and LiH molecule. In 2009, Wernli et al. [20] constructed a threedimensional PES for the electronic ground state of the reaction H + LiH ? H2 + Li by using the complete active space self consistent field (CASSCF) method followed by the multireference configuration interaction (MRCI) treatment of the correlation energy. The exothermicity of the reaction was estimated to be 2.258 eV [20], consistent with the value of 2.23 eV obtained by Dunne et al. [16]. The barrier to the reaction was also found to be absent on the PES of Wernli et al. [20]. The van der Waals well was found in the product valley of Li + H2 on the DMJ PES [16], but it was found to be absent on the PES of Wernli et al. [20]. Later that same year, Prudente et al. [21] also developed an accurate PES of the ground electronic state for the reaction H + LiH and performed the quantum wave packet calculations on it. Their calculations indicated that the Prudente PES clearly improved the DMJ PES [16] in several ways and was much smoother [21]. The dynamics of H + LiH collisions was also studied on the constructed PESs by the QM and QCT calculations [22–29]. In the recent past, a series of QM calculations on the DMJ PES [16] have been carried out to investigate the dynamics of this reaction in detail [22–27]. Defazio et al. [22] and Padmanaban and Mahapatra [23–27] applied the QM method to study the reaction probability, reaction cross section, thermal rate constant, mechanism and resonance of H + LiH scattering. Liu et al. [28] calculated the stereodynamics of the reaction H + LiH (v = 0, j = 0) ? H2 + Li by using QCT method based on the PES of Prudente et al. [21]. Moreover, Roy and Mahapatra [29] applied the time-dependent quantum wave packet dynamics method to study the initial state-selected channelspecific reactivity of H + LiH collisional system and its isotopic variants on the ab initio PES developed by Wernli et al. [20]. It was also worthy to note that the ionic variants of LiH2 system is the same important as the neutral system and shares many common features with the neutral reaction. The recent work on this subject was the quantum dynamics study of H + LiH+ reaction reported by Roy et al. [30]. They reported the channel-specific integral reaction cross-sections and thermal rate constants of the reaction H + LiH+. As mentioned above, most of the previous experimental and theoretical studies for the title reaction were focused on the rotational population distribution of LiH, reaction probability, cross section, rate constant, and resonance of H + LiH (v = 0, j = 0) collisions. However, there are few studies [25,29] related to the effects of the vibrational and rotational excitation of the reagent on the reactivity of H + LiH reaction. In this paper, we apply the QCT method to study the effects of the collision energy and vibrational excitation of the reagent on the reactivity of the title reaction on the Prudente PES [21]. We focus here on the dynamics of H + LiH reaction which can proceed via the following two reactive paths:
a standard Monte Carlo sampling of the initial conditions and the adaptive-step Adams method is used to integrate Hamilton’s equations. The conservations of both the total energy and total angular momentum have been checked carefully. In the calculation, the classical Hamilton’s equations are numerically integrated for motion in three dimensions, and the accuracy of the integration is verified by checking the conservation of the total energy and total angular momentum for every trajectory. In the present work, the vibrational and rotational levels of the reactant molecule are taken as m = 0–3, j = 0, respectively. Reaction probabilities and cross sections as a function of collision energy for the H + LiH reaction have been calculated by running a batch of 50,000 trajectories at the randomly and uniformly sampled collision energies within the range 0.001–1.0 eV. The trajectories are started at an initial distance of 15 Å between H atom and the center of the mass of LiH molecule. The total reaction probability Nr/Nt for total angular momentum J = 0 (the impact parameter equals to 0) is the ratio of the number of reactive trajectories Nr to the total number of trajectories Nt. In the QCT framework, for a fixed collision energy Ec and the given rovibrational state (v, j) of LiH, the reaction cross section rvj can be written as:
rv j ¼ pb2max ðNr =Nt Þ;
ð1Þ
where bmax denotes the maximum value of the reactive impact parameter for the collision energy Ec. 3. Results and discussion The QCT reaction probabilities for the LiH depletion (H + LiH (m = 0, j = 0) ? H2 + Li) and H-exchange (Ha + LiHb (m = 0, j = 0) ? LiHa + Hb) channels with J = 0 are displayed in Fig. 1. In order to certify the validity of the QCT method, we firstly compared the QCT probabilities with the previous quantum results, which can be also found in Fig. 1. The red line and blue line are quantum results reported by Prudente et al. [21] and Roy and Mahapatra [29], respectively. Prudente et al. [21] only calculated the reaction probabilities for collision energies between 0.1 and 1.0 eV. It is clear from Fig. 1 that the QCT results are in general agreement with the quantum results. The PES used by us is same with the one used by Prudente. The QCT results are slightly smaller than the QM results, it may be due to the lack of the quantum effects in QCT method. After certifying the effectiveness of the QCT method for this system, we investigate the effect of the reagent vibrational excitation
H þ LiH ! H2 þ Li ðR1Þ; Ha þ LiHb ! LiHa þ Hb ðR2Þ: Although the highly exothermic LiH depletion path (R1) is considered to contribute to LiH depletion, the thermoneutral hydrogen exchange path (R2) is thought to lead to the retention of LiH [23]. Besides, there is another non-reactive path leading to the retention of LiH. The paper is organized as follows: in Section 2 we briefly review the basic theory. In Section 3, results and discussion are presented. Finally, Section 4 gathers the conclusions. 2. Basic theory Details of the QCT method can be found in Refs. [31–34] and only a brief description is given here. The QCT calculations employ
Fig. 1. Comparison of the calculated H + LiH (m = 0, j = 0) reaction probabilities for total angular momentum J = 0 between the QCT (this work) and QM (cf. Fig. 3 of Ref. [21] and Fig. 1 of Ref. [29]) calculations of LiH depletion (R1) and hydrogen exchange (R2) channels.
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Fig. 2. QCT total reaction probabilities of the H + LiH (m, j = 0) reaction as a function of collision energy for various vibrational excitation states of the R1 and R2 channels.
on the reaction probabilities of R1 and R2 channels. The J = 0 total reaction probabilities for the various vibrational quantum state of LiH molecule (m = 0–3) are plotted as a function of collision energies and the results are presented in Fig. 2. It can be seen from Fig. 2 that the vibrational excitation of the reagent LiH causes a substantial reduction of the probabilities of R1 channel, whereas it causes a substantial increase of the probabilities of the R2 channel. The extent of decrease and increase of the two channels are quite different between v = 0 and v – 0. In addition, it is observed that the exothermic LiH depletion reaction is more efficient than the thermoneutral H-exchange reaction. Similar findings were also reported by Wernli et al. [20], Prudente et al. [21] and Roy and Mahapatra [29]. The QCT integral cross sections of the reaction H + LiH (m, j = 0) for the R1 and R2 channels are presented in Figs. 3a and 4a. The reaction cross sections for the various vibrational quantum state of LiH molecule (m = 0–3) are plotted as a function of collision energies. In the given energy range, Roy and Mahapatra [29] accurately calculated the partial wave contribution for total angular momentum up to J = 70. They found that J = 42 is necessary to obtain the
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converged cross sections for the R1 and R2 reactive channels, respectively. We find that the reaction cannot be carried out below the collision energy of 0.001 eV and begins to happen at about 0.002 eV. It means that the PES of LiH2 system exists a very small barrier, which causes a nonzero threshold in the reaction H + LiH. It is important to note that Liu et al. [28] found a small barrier at the collinear arrangements of LiH2 system. The PES used by us is same with the one used by Liu et al. A very small barrier appears along the path for the LiH depletion reaction, which is basically consistent with the barrier-less description in Ref. [15]. It can be seen from Fig. 3a that the QCT cross sections decrease with the increase of the collision energy, as expected for a barrier-less highly exothermic reaction. Moreover, it is clear from Fig. 3a that the vibrational excitation of the reagent LiH leads to a substantial reduction of its depletion cross section, whereas it results in a substantial increase of its H-exchange reactivity [cf. Fig. 4a]. Namely, the vibrational excitation of the reagent LiH has a negative effect on the reactivity of the LiH depletion channel, whereas it promotes the reactivity of the H-exchange channel. These trends are consistent with the earlier findings [25,29] (cf. Fig. 7 of Ref. [25] and Fig. 4 of Ref. [29]) on the PESs from DMJ and Wernli et al. Polanyi and coworkers [35] have extensively studied the effects of the reagent vibrational and rotational excitation on the reactivity of a chemical reaction system. Generally, the vibrational excitation of the reagent can promote the reactivity of R1 channel while the PES exists a late barrier [35]. The minimum energy paths (MEPs) of the collinear geometry for R1 channel of the title reaction is attractive [28] and no barrier is found in the exit valley [21]. Therefore, it is reasonable that the vibrational excitation of the reagent inhibits the reactivity of the depletion channel. The dynamics of the R2 reactive channel seems to be more complicated. It has been showed that the R2 channel exists the potential basin along the collinear minimum energy path, which appears at the distance of the closest approach of the reagents and is much deeper on the DMJ PES [16]. The potential basin permits an effective transfer of the translational and vibrational energies to the R2 reactive channel [36], therefore, the reagent approaches each other and further to bind together at the distance of the closest approach. Similar findings were also reported by Roy and Mahapatra [29]. From Figs. 3a and 4a, we can also conclude that the effects of the reagent vibrational excitation on the cross sections seems to be more marked at the
Fig. 3. The reaction cross section of the reaction H + LiH (m = 0–3, j = 0) ? H2 + Li as a function of collision energy for the QCT results from this work are compared with QM results of Ref. [29].
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Fig. 4. The reaction cross section of the reaction Ha + LiHb (m = 0–3, j = 0) ? LiHa + Hb as a function of collision energy for the QCT results from this work are compared with QM results of Ref. [29].
lower collision energy than at the higher collision energy. The QM results of Roy and Mahapatra [29] are also shown for comparison in Figs. 3b and 4b. When the values of QCT depletion cross sections are expanded 1.5 times, the shape of the present QCT reaction cross sections is very similar to that of Roy et al. although the magnitude
Fig. 5. Representative reactive trajectories for R1 for LiH vibration states
of the reaction cross sections is somewhat smaller than the Roy’s result. These differences are probably as a result of the different PESs and the lack of the quantum tunneling in the QCT method. But overall, the trend of the present QCT results are in good agreement with the available QM results.
v = 0–3, as plotted on the collinear HHLi surface at the collision energy of 0.05 eV and ht = 43°.
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Fig. 6. Representative reactive trajectories for R1 for LiH vibration states
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v = 0–3, as plotted on the collinear HHLi surface at the collision energy of 0.2 eV and ht = 69°.
The reactivity of the reaction H + LiH can be explained by examining the representative reactive trajectories for the different vibrational excitation, and the topology of the HHLi PES as shown in Figs. 5 and 6. To obtain a clearer understanding of the reaction mechanism, we observe lots of representative reactive trajectories which plot on the collinear HHLi surface. The contours of PES are constructed as the function of RHH and RLiH. It is worth noting that we find a very small barrier for the collinear path and it is perfectly consistent with the observations of Liu et al. [28]. The trajectories of the depletion channel with the variation of internuclear distances Ha–Hb and Hb–Li for the H + LiH (m, j = 0) reaction, which plot on the collinear HHLi surface, are also presented in Figs. 5 and 6. The trajectories are selected in scattering angle ranges of ht = 43° and 69° at the lower collision energy of 0.05 and 0.2 eV. In Fig. 5a, the initial state selected trajectories of the reaction H + LiH (m = 0, j = 0) show an interesting phenomenon at the collision energy of 0.05 eV. We observe many indirect reactive trajectories, which are called ‘‘trapped’’ trajectories. Although the PESs of DMJ [16] and Prudente et al. [21] are both highly attractive along the collinear minimum energy path of the LiH depletion channel, the DMJ PES supports two unphysical wells, which appear in the reagent and product channels along the minimum energy path on the DMJ PES [16,29]. Owing to the Prudente PES [21] was achieved by performing accurate ab initio full configuration interaction calculations and fitting them to a slightly modified version of the analytical functions of DMJ [16], the existence of unphysical wells in the entrance or exit valleys make the longlived trajectories emerge on both PESs. It is clear from Fig. 5a that the projectile atom H undergoes several collisions with the target molecule LiH and the complexes have relatively long lifetimes
before breaking-up into one of the random channels. The relatively long-lived survival of the metastable states could permit the system to perform the extensive rotations before breaking up, therefore make the products appear in the more random directions. Fig. 5b and c present a non-reactive trajectories, which implies that the vibrational excitation of the reagent decreases the reactivity of the R1 channel. The results reasonably explain why the vibrational excitation of the reagent causes a substantial reduction of cross section of this system. To our surprise, in Fig. 5d we observe the H-exchange channel trajectory of the reaction H + LiH (for initial m = 3, j = 0) at the collision energy of 0.05 eV. In other words, the vibrational excitation of the reagent LiH causes a substantial reduction of its depletion reactivity, whereas it causes a substantial increase of its exchange reactivity. Fig. 6a and b show that the H atom collides with the LiH molecule and forms a H2 molecule that moves away immediately at the collision energy of 0.2 eV and ht = 69°. This process is in conformity with the dynamics of the direct reaction. Besides, at the higher collision energy range, we do not find any trajectories of the indirect reaction. We only show the representative trajectories at Ec = 0.05 and 0.2 eV here. In most cases, the product molecule H2 moves away immediately when the attacking atom H collides with the LiH molecule, and the collision process is relatively short-lived. Overall, the H + LiH collisions is mainly dominated by the direct reaction mechanism, but the indirect mechanism plays a role while the collision energy is lower. As can be seen in Fig. 6c and d, the attacking atom Ha collides with the LiHb molecule and forms a LiHa molecule that moves away immediately. That is to say, the vibrational excitation of the reagent plays an important role on the reactivity of the reaction H + LiH. It also illustrates that the vibrational excitation of the reagent
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LiH inhibits the reactivity of the depletion channel, whereas it obviously promotes the reactivity of the H-exchange channel. The corresponding QCT results for the initial state m = 0–3 and j = 0 are in good agreement with the available QM results from the recent literature [29]. 4. Conclusion In this paper, QCT calculations have been performed to study the effects of the collision energy and the vibrational excitation of the reagent on the reactivity of H + LiH reaction. We first calculate the J = 0 total reaction probabilities of both the LiH depletion channel and H-exchange channel, and the QCT results are compared with the recent available literature data. The reactivity of the LiH depletion channel is found to be greater than the H-exchange channel, which is very similar to the reaction H + LiH+ reported by Roy et al. [30]. Meanwhile, the present QCT results are in good accord with the available QM results from the recent literature. On this basis, we investigate the influence of the collision energy and the vibrational excitation of the reagent on the reaction cross sections of the H + LiH reaction. The reaction cross sections decrease with increasing collision energy. While we find that if v – 0 and j = 0, the vibrational excitation of the reagent LiH decreases the reactivity of the depletion channel and increases the reactivity of the H-exchange channel. Such a trend has been also pointed out in a recent work [29]. This phenomenon can be explained by examining the representative reactive trajectories for the different vibrational excitation on the collinear HHLi surface. The calculated results imply that the collision energy and the vibrational excitation of the reagent play an important role on the reactivity of this system. It is important to note that the above findings also have further confirmed the reliability of the results obtained by Roy and Mahapatra [29]. Acknowledgements The authors are very grateful to Professor Frederico V. Prudente for providing the codes of the potential energy surface [21]. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11074103, 10974078, and 11174117) and Discipline Construction Fund of Ludong University. All calculations were carried out in the Shuguang Super Computer Center (SSCC) of Ludong University. The authors also appreciate Professor Han for providing the QCT code. References [1] S. Lepp, J. Shull, Molecules in the early universe, Astrophys. J. 280 (1984) 465– 469. [2] N. Yoshida, T. Abel, L. Hernquist, N.S. Yoshida, Simulations of early structure formation: primordial gas clouds, Astrophys. J. 592 (2003) 645–663. [3] T. Prodanovic´, B.D. Fields, On nonprimordial deuterium production by accelerated particles, Astrophys. J. 597 (2003) 48–56. [4] T. Prodanovic, B.D. Fields, Probing primordial and pre-galactic lithium with high-velocity clouds, Astrophys. J. 616 (2004) L115–L118. [5] S. Lepp, P.C. Stancil, A. Dalgarno, Atomic and molecular processes in the early Universe, J. Phys. B: At., Mol. Opt. Phys. 35 (2002) R57–R80. [6] E. Bodo, F.A. Gianturco, R. Martinazzo, The gas-phase lithium chemistry in the early universe: elementary processes, interaction forces and quantum dynamics, Phys. Rep. 384 (2003) 85–119. [7] D. Galli, F. Palla, The chemistry of the early Universe, Astron. Astrophys. 335 (1998) 403–420.
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