Mechanism of the Collision Energy and Reagent Vibration's effects on the Collision Time of the Reaction Ca + HCl

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Mechanism of the collision energy and reagent vibration’s effects

on the collision time for the reaction Ca + HCl

Xiaohu He

a,b,c

_{, Victor Wei-Keh Chao (Wu)}

d

_{, Keli Han}

a

_{, Ce Hao}

b

_{, Yan Zhang}

a,⇑

a

State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, China

b

School of Chemical Engineering, Dalian University of Technology, Dalian 116023, China

c

University of Chinese Academy of Sciences, Beijing, China

d

Department of Chemical and Materials Engineering, National Kaohsiung University of Applied Sciences, Jian-Gong Road, San-Ming Section, Kaohsiung 80782, Taiwan

a r t i c l e

i n f o

Article history:

Received 27 November 2014

Received in revised form 25 December 2014 Accepted 26 December 2014

Available online 12 January 2015 Keywords:

Average collision time Ca + HCl reaction Attack angle

Collision time product distribution

a b s t r a c t

The collision time which describes the speed of the collision process in a reaction is an important concept to an elementary chemical reaction. In this study, the quasiclassical trajectory method is applied to inves-tigate the collision time of the reaction Ca + HCl (v = 0–2, j = 0) ? CaCl + H. In order to provide a clear image of the reaction, the integral cross section we calculated is compared with corresponding quantum result and shows fairly good agreement. The results indicate that the collision energy and the initial vibrational level affect the average collision time remarkably. As the collision energy or the initial vibrational level increases, the average collision time decreases. The difference of average collision time for different initial vibrational level decreases with the increasing of collision energy. The product distri-butions as functions of scattering angle, attack angle and impact parameter are computed. Observing the functions, it can be found that the features could be caused by a competition among different parts of the product molecules with different collision time. For all the investigated initial vibrational levels, most of the reactive trajectories have the shorter collision times and are focused in several concentrated regions. Two possible mechanisms could be responsible for the HCl (v = 0) reaction in the concentrated regions. One is the sideway scattering and the system would fall into the deep potential well once in the collision process. The other is the weak forward scattering and strong backward scattering. The system would go around the deep potential well in the collision process. It is shown that the character of the weak forward scattering and strong backward scattering for the HCl (v = 1 and 2) reactions in the concentrated regions. However, the reactions outside the concentrated regions have the longer collision times and no particular mechanism. In the collision process, the system could fall into the deep potential well many times. We also explored the dynamics of the reaction at the same total energies but for different initial vibrational levels and found that the role of the insertion well becomes less and less important with the increasing of total energy.

1. Introduction

In past decades, chemical reactions between metal atoms and halide molecules have been the subject of theoretical and experi-mental studies. Generally the systems consisting of alkali (or alkali-earth) metal atoms (M) and hydrogen halides (HR, which R is a halide element such as F, Cl, Br or I) are the most representative prototypes of these reaction systems. These systems may be appropriate models to study the effects of collision energy and ini-tial quantum state of the reagents on the reaction dynamics. These reactions are mainly dominated by the harpoon mechanism[1–3].

In the mechanism, one electron (or several electrons) of the metal atom ‘‘jumps’’ to the HR molecule and forms an anion. Then the anion splits to product molecule MR and H. Mestdagh et al. reviewed these reaction systems and gave a comprehensive description of them[1].

The molecular reaction dynamics for several hydrogen halides reacting with several alkali metal atoms (Li, Na)[2,4–7]and some reactions involving several alkali-earth metal atoms (Ca, Be, Sr, Mg, Ba) [8–19] have been theoretically and experimentally investi-gated. In the reactions with hydrogen halides, there is a similarity between alkali and alkali-earth metal atoms. The reaction process can be revealed as some crossings of potential curves between covalent states of the reagents and ionic states of the products

[3,20]. Also, there is a vital difference between these two kinds of

http://dx.doi.org/10.1016/j.comptc.2014.12.023

⇑ Corresponding author. Tel.: +86 411 84379483; fax: +86 411 84675584. E-mail address:zhangyanhg@dicp.ac.cn(Y. Zhang).

Computational and Theoretical Chemistry 1056 (2015) 1–10

Contents lists available atScienceDirect

Computational and Theoretical Chemistry

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atoms[3,20]. An alkali metal atom has only one active electron, so one crossing is usually enough to understand the reaction process

[21,22]. For an alkali-earth atom, it has more than one active elec-tron. Thus, there are several crossing appearances which lead to multiple harpoon mechanisms (see Ref. [23]and the references therein). The reactive collision between alkali-earth metal atoms and hydrogen halides contains complicated variation in the electronic structure, which leads to various dynamics behaviors under different conditions. Moreover, the divalent nature of the alkali-earth metal atoms leads to a deep insertion well which cor-responds to a collinear H–M–R complex. The deep well may be the main factor that contributes to the resonances with very long life time[20].

One of the widely studied systems in the reactions of an alkali-earth atom and a hydrogen halide is Ca + HCl ? CaCl + H, and many of the studies focused on the reactions with excited Ca atoms (see Ref.[3]and the references therein). The Ca (1_{S) + HCl van der} Waals complex had been studied experimentally[24–26]and the formation of the product CaCl was observed at particular electronic states[26]. It had been verified that the van der Waals complex Ca–HCl plays a vital role in the Ca + HCl reaction[24,27,28]. More-over, Lawruszczuk et al. explained the importance of the ground electronic state of the system[26]. In this paper only the computa-tional results of the ground state potential energy surface (PES) of the system were concerned. Theoretically, it is rather difficult to obtain sufficiently accurate PES for this system, because of the large amount of electrons in the system and complicated electronic structure change on the reaction path[3,20]. Before the accurate ab initio PES of the Ca + HCl reaction was reported by Verbockha-ven et al.[3], there were only several PESs applied the diatomics-in-molecules (DIM) method[29,30]. In this work, all our reaction dynamics calculations are based on the PES of Verbockhaven et al.[3].

The PES developed by Verbockhaven et al.[3]is an accurate and full-dimensional PES of the Ca + HCl system on the ground state. It was obtained from a series of strict ab initio calculations at the multireference conﬁguration-interaction (MRCI) level. The reaction Ca + HCl is endothermic by about 0.63 eV. Along the minimum energy reaction path, there is an energy barrier of about 0.55 eV. On both sides of the barrier, two energy wells appear at linear geometries. On the entrance channel it is a very shallow van der Waals well of about 0.024 eV, which is much smaller than the insertion potential energy well (about 2.09 eV) attributed to the linear H_Ca++_Cl _{conﬁguration}_[3]_{. The reaction mechanism can} be explained by the competition between the direct reaction which is dominated by the energy barrier and the indirect reaction which is attributed to the energy wells.

Theoretical studies about the Ca + HCl reaction’s dynamics are quite scarce. Sanz et al. ﬁrstly studied this reaction for different ini-tial quantum states of the HCl reagent using time-dependent wave-packet (TDWP) method, and they also simulated the infrared excitation from the Ca–HCl van der Waals well[20]. They found that the reaction is essentially direct, and there is no obvious change in the reaction efﬁciency as a function of the initial HCl quantum state. In addition, the reaction has an energy threshold at about 0.5 eV. Tang et al. studied the product polarization distri-bution of the Ca + HCl reaction at the collision energy of 20 kcal/ mol, as well as the product ro-vibrational state distribution[31]. Wang et al. carried out a series of studies on the stereodynamics of the Ca + HCl reaction[32–34]. They have studied the effects of isotopic variant [32], vibrational excitation of the HCl reagent

[33] and collision energy [34] on the stereodynamics of this reaction.

Although the aforementioned investigations fully covered the collisional dynamics and stereodynamics, we think it is interesting and necessary (or even inspiring) to explore how the reaction is

shown on the dimensionality of time. The application of the con-cept of collision time[35]allows us to study the time evolution of the reaction process, as well as the speed of the reaction. In this work the collision time of the reaction are studied under different initial vibrational state of the reagent HCl and a wide range of col-lision energy (0.1–2.0 eV). The calculated colcol-lision time results are exhibited and discussed in detail related to different dynamics properties, including scattering angle, attack angle[36]and impact parameter. The paper is arranged as follows. In Section 2, we brieﬂy introduce the methodology and computational details applied here, and all the results are presented and discussed in Sec-tion3. In Section4we summarized the main conclusions of this work.

2. Theory and computation details

The QCT calculation method is basically the same as previous works[37–42]. Only some key details are presented here. The motions of the three atoms are simulated by a series of Hamilton’s equations which are integrated with six order symplectic integration[43]. The collision energy Ecis chosen in the range of 0.1–2.0 eV, and the interval of two neighboring energies is 0.1 eV. The initial vibrational quantum number of reagent HCl is set as v = 0–2. The initial rotational quantum number j is 0. At least 1.5 105 _{trajectories were run for each single condition (E}

c, v and j). The integration step size is chosen as 0.1 fs which guaran-teed the calculated accuracy of the total energy and total angular momentum conservation better than 104_{and 10}6_{, respectively.} The conservation is scaled by the equation |Vi Vf|/Vi, where Vi and Vfare the initial and ﬁnal values (total energy or total angular momentum), respectively. The initial distance from the atom Ca to the CM of the diatom HCl is set to 12 Å to ensure negligible initial interaction between them. The maximum impact parameters are properly set according to the initial condition of calculation.

The collision time

s

colis calculated by the following equation

[35]:

s

col¼

s

tot R0 Rint

v

r R0 0 Rint

v

0 r : ð1Þ

s

totis the total duration time of a trajectory. R0and R00are the initial and ﬁnal atom–diatom distances, respectively.

v

rand

v

0r are the initial and final relative velocities, respectively. The truncation dis-tance Rintis set to 4.2 Å. According to the definition, the collision starts at R0= Rint, and ends at R00¼ Rint. Thus, the truncation distance Rintdefines the boundary of the collision or interaction. The trunca-tion distance is determined by carefully investigating the potential energy variation as a function of the reagent and product atom–dia-tom distances on the condition that all collision energies and initial vibrational levels are considered. In the frame of this definition the collision time is always greater than zero, even for the most direct trajectory. The definition shows that the collision time is also inde-pendent of initial and final states of the system. Hence, a common time concerning only the drastic interaction in the interaction region is allowed in all cases. The calculation process of the collision time applied in this work has been carefully tested and shows favorable numerical stability.

The average collision time

s

avgis calculated as

s

avg¼

PNr i¼1

s

col;i

Nr

; ð2Þ

where

s

col,iis the collision time of i th trajectory, and Nris the num-ber of reactive trajectories.

The ﬁnal product distributions as functions of collision time, scattering angle, attack angle and impact parameter are repro-duced by a ﬁtting process using the method of moment expansion

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in Legendre polynomials (see Refs. [44,45] and the references therein). The main character is briefed as follows. If we assume x (1 6 x 6 1) to be a reduced variable, and G(x) its probability density distribution function which is normalized to one, the expression of G(x) in series of Legendre polynomials will be:

GðxÞ X

M

n¼0

anPnðxÞ: ð3Þ

The expansion is truncated at the M th term. The coefﬁcients anare calculated as an¼ 2n þ 1 2 PNa i¼1PnðxiÞ Na ; ð4Þ

where Nais the number of reactive trajectories in ﬁnal state

a

.

The product distribution as a function of collision time

s

colis calculated as

Pð

s

colÞ ¼ bGð

s

Þ: ð5Þ

s

is a reduced variable which is deﬁned as

s

= 2

s

col/

s

max 1. P(

s

col) is expanded up to M = 120 and shows rational convergence. b is adjusted to make the maximum value of P(

s

col) is equal to one in order to bring out the distinction of the collision time product distributions between different collision energies and conceal the variation caused by the reaction probability.

The product distribution as a function of the scattering angle ht (the differential cross-section (DCS)) is calculated as

PðhtÞ ¼

r

R

2

p

Gðcos htÞ; ð6Þ

where

r

Ris the integral cross section. P(ht) describes the product probability density distribution in the scattering angle.

In this paper, we introduce the deﬁnition of the attack angle

[36]and show it inFig. 1schematically. As shown inFig. 1, the attack angle is the angle between the line of atom A to CM point (the center-of-mass of BC molecule) and the line of CM point to atom B in an A + BC ? AB + C reaction. The attack angle is actually the supplementary angle to the initial angle in Jacobi coordinate. The initial Jacobi angle is symboled as

c

in the ﬁgure in order to make the attack angle prominent.

The calculation of product distribution as a function of attack angle h is similar to P(ht):

PðhÞ ¼

r

R

2

p

Gðcos hÞ: ð7Þ

P(h) describes the product probability density distribution in attack angle.

The product distribution as a function of impact parameter b is calculated as

PðbÞ ¼ 2

r

R

p

b2max

GðxÞ; ð8Þ

where x ¼ 2b2=b2max 1. bmaxis the maximum impact parameter. P(ht), P(h) and P(b) are expended up to M = 30 and show good convergence. P(ht), P(h) and P(b) are all product probability density functions representing speciﬁc product distributions in some phys-ical quantities, i.e., scattering angle, attack angle and impact parameter in these cases.

3. Results and discussion

In order to give an overall and clear image of the Ca + HCl reac-tion, the discussion is started with the integral cross section (ICS) result, which is portrayed inFig. 2. The quantum result[20]is also portrayed in the same ﬁgure. The classical ICS is calculated by the equation

r

R¼

p

b

2

maxNR=NT, where NRand NTare the numbers of reactive and total trajectories, respectively. The number of total trajectories is very large, thus the statistical error is negligible (less than 1% for all conditions). Observing the figure, one can find that for the ground initial vibrational state, the ICS almost linearly increases with the increasing of collision energy, and shows an energy threshold at about 0.5 eV. For the excited initial vibrational states, the ICSs first increase with the increasing of collision energy, then become insensitive to the collision energy. For v = 1 reaction, the agreement between classical and quantum results is surpris-ingly good, although there is a small gap between the locations of two thresholds. In contrast, for v = 0 reaction the locations of the thresholds are almost the same, but the magnitudes of the ICSs apparently differ from each other. This difference may be due to that classical method is not capable to describe the resonance phenomenon which is attributed to the deep insertion well of the CaHCl system.[20]In general, the ICSs shown inFig. 2could reveal an important character of this reaction, which is that the energy barrier plays a vital role in the reaction.

The average collision time

s

avgof a reaction at a speciﬁc initial condition is deﬁned as the average value of each reactive trajec-tory’s collision time. From

s

avgone can easily tell how fast the col-lision process of the reaction is. As shown inFig. 2,

s

avgdecreases with the increasing of collision energy for all the three vibrational states. It seems that the increasing of the collision energy lessens the running time which the complex would spend in the potential energy well. It has been reported that some resonances which is due to the insertion potential well appear at low collision energies (especially around the reaction threshold), and much less reso-nances appear at high collision energies[20]. Resonances can be attributed to the insertion well, which is expressed by long-life time complex with different

s

avg at different collision energies

Fig. 1. Schematic diagram of the attack angle. The attack angle is the angle between line A–CM and line CM–B, which is just the supplementary angle of the traditional Jacobi anglec. 0.0 0.4 0.8 1.2 1.6 2.0 0 4 8 12 16 20

Integral cross section (Å

2) v = 0 v = 1 v = 2 v = 0 Roncero et al. v = 1 Roncero et al.

Collision energy (eV)

Fig. 2. Integral cross sections for the Ca + HCl (v = 0, 1 and 2) reactions. The present results are compared with previous quantum results. The quantum results are taken from Ref.[20].

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energy the reaction is mainly led by mechanism 2. ObservingFig. 9

and reviewing the discussions uponTables 1 and 2andFigs. 4 and 8, it is reasonable to conclude that the role of the insertion well in the reaction is less and less important with the increasing total energy. It has been discussed that the initial vibrational excitation of the HCl molecule may yield to products with high kinetic ener-gies [20], explained that the ability of the system to avoid the insertion well increases with the total energy.

4. Conclusion

In this work, a series of QCT calculations have been carried out to study the effect of collision energy and reagent vibrational exci-tation on the collision time of the title reaction. In order to provide an overall and clear image of the reaction, the integral cross sec-tions (ICSs) at different initial condisec-tions are also presented, which shows fairly good agreement with the quantum ICSs. Scattering angle, attack angle and impact parameter are associated to the col-lision time in order to analyze and explain the reaction mechanism. It has been found in our calculations that both the collision energy and reagent vibrational excitation signiﬁcantly decrease the aver-age collision time, i.e., accelerate the reaction.

For all the investigated vibrational states, the reaction is mainly dominated by the deep insertion potential well. Especially, the product molecules show an almost symmetric forward–backward scattering at the low collision energies around the threshold. At rel-atively high collision energies, the reactivity is focused on some concentrated regions. The reaction of the ground initial vibrational state can be described by a competition between two concentrated regions. Product molecules of region 1 are mainly scattered side-way, and arise from the collisions at the attack angles less than 120°. In contrast, product molecules of region 2 show the weak for-ward scattering and strong backfor-ward scattering, and they almost arise from the collisions at small and large attack angles (around 0° and 180°). It is also found that the product molecules of the concentrated regions mainly arise from small impact parameter collisions comparing the global product distribution. For the two excited vibrational states the reaction mechanism of the concen-trated regions is coincident with that of the weak forward scatter-ing and strong backward scatterscatter-ing for the ground vibrational state. Observing the variation of potential energy and internuclear distances, we obtained two types of reactive trajectories for the Ca + HCl reaction. The two-type reactive trajectories give the two possible mechanisms of the collision time for the Ca + HCl reaction.

In the ﬁrst case, the system falls into the deep potential well once and it is responsible for the region 1 of v = 0. In the last case the system does not fall into the deep potential well but gets around it in the collision process and this describes the mechanism of region 2 for v = 0 and the concentrated regions for v = 1 and 2. Trajectories outside the concentrated regions have much longer collision time and have no particular dynamics information. In those regions, the system could go into and escape from the deep potential well many times.

Moreover, we explored the dynamics of the reaction at several total energies for different initial vibrational states, and we came to the conclusion that the role of the insertion well becomes less and less important with the increasing of total energy.

Acknowledgment

This research was supported by the National Natural Science Foundation of China No. 21103167.

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