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Accurate Quantum Wave Packet Study of the Deep Well D+ + HD Reaction: Product Ro-vibrational State-resolved Integral and Differential cross sections Haixiang He, Weimin Zhu, Wenli Su, Lihui Dong, and Bin Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b08941 • Publication Date (Web): 19 Feb 2018 Downloaded from http://pubs.acs.org on February 19, 2018

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The Journal of Physical Chemistry

Accurate Quantum Wave Packet Study of the Deep Well D+ + HD Reaction: Product Ro-vibrational State-resolved Integral and Differential cross sections Haixiang He*, Weimin Zhu, WenLi Su, LiHui Dong,andBin Li School of Chemistry & Chemical Engineering, Guangxi University, Nanning 530004, Guangxi, People’s Republic of China KEYWORDS: quantum wave packet, integral cross section, differential cross section

ABSTRACT: The H+ + H2 reaction and its isotopic variants as the simplest triatomic ionmolecule reactive system have been attracting much interests, however there are few studies on the titled reaction at state-to-state level until recent years. In this work, accurate state-to-state quantum dynamics studies of the titled reaction have been carried out by a reactant Jacobi coordinate-based time-dependent wave packet approach on diabatic potential energy surfaces constructed by Kamisaka et al. Product ro-vibrational state-resolved information has been calculated for collision energies up to 0.2 eV with maximal total angular momentum J = 40. The necessity of including all K-component for accounting the Coriolis coupling for the reaction has

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been illuminated. Competitions between the two product channels, (D+ + HD'→ D'+ + HD and D+ + HD'→ H+ + DD') were investigated. Total integral cross sections suggest that resonances enhance the reactivity of channel D+ + HD'→ H+ + DD', however, resonances depress the reactivity of the another channel D+ + HD'→ D'+ + HD. The structures of the differential cross sections are complicated and depend strongly on collision energies of the two channels and also on the product rotational states. All of the product ro-vibrational state-resolved differential cross sections for this reaction do not exhibit rigorous backward-forward symmetry which may indicate that the lifetimes of the intermediate resonance complexes should not be that long. The dynamical observables of this deuterated isotopic reaction are quite different from the reaction of H+ + H2→ H2 + H+ reported previously.

I. INTRODUCTION As the simplest triatomic ion-molecule reactive system with only two electrons, the H+ + H2 reaction and its isotopic variants, which serves as a paradigm and plays an important role in interstellar chemistry, has been attracting many experimental and theoretical interests and have been extensively investigated during the last decades [1-3]. Due to deep potential well, longrange potentials and strong nonadiabatic effects associated with charge-transfer in a typical ionmolecule reaction, it is still quite challenging for accurate quantum dynamics calculations for the reaction of H+ + H2 and its isotopes at the state-to-state level. Till now, quite several adiabatic and diabatic potential energy surfaces (PESs) for describing the titled reaction have been developed and a number of corresponding dynamics studies of the reaction have been reported.


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In 1996, Ichihara et al. [4] calculated the cross sections for the D+ + H2, D+ + D2 and H+ + D2 reactions on their ab initio PES [5] using the trajectory-surface-hopping method. The comparison between their theoretical results and the experimental data suggested the necessity of performing more accurate quantum calculations for this reaction. Soon later, an adiabatic global PES for the H3+ system has been reported using hyperspherical coordinates for the ground state [6] and the lowest triplet state [7]. Based on the adiabatic PESs, dynamic studies using various theoretical methods were carried out. Total reactive probability, integral and differential cross sections (ICSs and DCSs) have been calculated for collision energies lower than ~1.5 eV, which is just below the electronic curve crossing energy. The theoretical results were compared with the available experiments [8-10] for the H+ + H2, H+ + D2 and D+ + H2 reactions. H+ + H2 reaction involves two or more potential energy surfaces and electronically nonadiabatic transitions occur at such crossing regions, which impels the development of more accurate and analytic diabatic potential energy surfaces. In 2002, Kamisaka et al. [10] reported an analytic diabatic PES (KBNN PES) based on their ab initio energy points reported in 1995. At the same time, preliminary three-dimensional quantum dynamic calculations for the D+ + H2 reaction have been carried out only for J = 0 for six adiabatic and nonadiabatic processes. Since the availability of analytic KBNN PES, by using the more computationally economic timedependent quantum wave packet (TDWP) approach, Han and coworkers [11] reported accurate quantum dynamic investigations for the total ICS and rate constants for the reaction of D+ + H2, H+ + D2 and other isotopes, where they also discussed the necessity of rigorous consideration for the coupled-channel (CC) calculations for the reactive charge transfer (RCT), the nonreactive charge transfer (NRCT), and the reactive noncharge transfer (RNCT) channels [12,13]. Recently, Hankel et al. [14] also investigated the effects of the Coriolis coupling for the H+ + D2 reaction at



low collision energies but using the adiabatic PES. Their calculations showed the truncated CC approach works well in certain cases and the full CC approach has also been proved to be imperative in quantum dynamics calculations for this reaction. Adiabatic dynamics for the two reactions, D+ + H2, H+ + D2, and their reverse processes have been studied by Jambrina et al. [1517], where they compared the results from several theoretical approaches, such as quantum wavepacket method, quasi-classical trajectory and statistical quasi-classical trajectory. They also made comparison between their theoretical results and the experimental data. In the literatures, besides the theoretical studies mentioned above, the dynamics of the titled reaction and its isotopes have also been investigated theoretically by many others [18-22], based on the PES by Aguado et al. [6] or Kamisaka et al. [10]. Despite of many studies on the H+ + H2 reaction and its isotopic variants, there are few studies on the titled reaction at the state-to-state level. The purpose of the present work is to study the reaction by a reactant coordinate-based state-to-state quantum wave-packet method for the processes D+ + HD'（v = 0, j = 0） → D'+ + HD (channel 1) → H+ + DD' (channel 2) with collision energy regime from 0.01 eV up to 0.4 eV based on the KBNN PES. Since the threshold energy for the opening of the charge transfer channel is much higher than the collision energies we considered, the nonadiabatic processes including the RCT and NRCT channels can be safely neglected thus only the reactive pathways of the hydrogen and deuterium atom exchanges need to be considered.


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The article is organized as follows. Section II outlines the theory of TDWP method for the dynamics studies. The CC method and truncate CC approximation are both utilized in the present calculations and the latter serves as a test for the validity of CC for the titled reaction. The calculated results and discussion are given in section III. The conclusion is presented in section IV.

II. METHODOLOGY Essentially, application of the TDWP method to study the triatomic A + BC → AB + C, AC + B reaction at the state-to-state level is to solve the time-dependent Schrödinger equation:

ih

∂Ψ ˆ = HΨ ∂t

(1)

In the reactant Jacobi coordinate, the Hamiltonian for a given total angular momentum J can be written as [23]

h2 ∂ 2 (J − j )2 j2 Hˆ = − + + + V ( R, r ) + hˆ(r ) 2µ R ∂R 2 2µ R R 2 2µr r 2

(2)

where µ R is the reduced mass between the center of mass of A and BC, J is the total angular momentum operator, j is the rotational angular momentum operator of BC, and µ r is the reduced mass of BC. The diatomic reference Hamiltonian is defined as

h2 ∂ 2 ˆ h(r ) = − + V (r ) 2µr ∂r 2


(3)


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where V(r) is the diatomic reference potential. The initial wave packet is expanded in terms of the body-fixed (BF) translational-vibrational-rotational basis [23-24]: ε ΨνJM ( R, r , t ) = 0 j0 K 0

F ν εν ∑ ν JM n jK ,

0 j0 K 0

(t )uνn ( R)φν ( r )Y jKJMε ( R, r )

(4)

n, , j , K

Here, uνn (R) is the translational basis function for R, φν (r ) is the eigenfunction for hˆ(r ) given in Eq. 3, and Y jKJMε is the total angular momentum eigenfunction in the BF representation where n and ν are the indices labeling the translational and vibrational eigenfunctions, respectively, and M and K are the projection quantum numbers of J on the space-fixed z-axis and BF z-axis, respectively, and ε is the parity of the system. Further, the Y jKJMε ( R, r ) in the BF representation can be written as [24]

Y jKJMε ( R, r ) = (1 + δ K 0 ) −1 / 2

[

]

2J + 1 J DK , M + ε (−1) J + K D−JK , M Y jK (θ ,0) 8π

(5)

where, DKJ , M is the Wigner rotation matrix, and Y jK are the spherical harmonics. On this basis, the element of the centrifugal term or the Coriolis coupling matrix is expressed as [25] h2 2µR R

Y

JMε jK

{[

]

h2 2 JMε ˆ ˆ δ jj ' J ( J + 1) + j ( j + 1) − 2 K 2 δ KK ' ( J − j) Yj'K ' = 2µ R R + JK

+ jK

− JK

− jK

− λ λ (1 + δ K 0 ) δ K +1, K ' − λ λ (1 + δ K 1 ) δ K −1, K ' 1/ 2

1/ 2

}

(6)

and λ is defined as

λ±AB = [ A( A + 1) − B( B ± 1)]1 / 2


(7)

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In the centrifugal-sudden (CS) approximation, the off-diagonal elements are neglected [26, 27]. In the calculations with initial rotational state as j = 0, it is convenient to construct an initial wavepacket in the space-fixed representation which adopts simple Gaussian function as [23-24]

Ψ (0) = G ( R)φν 0 k 0 (r )Y jKJMε ( R, r )

(8)

1/ 4

 1  G ( R) =  2   πδ 

exp(−( R − R0 ) 2 / 2δ 2 )e − ik 0 R

(9)

Then we applied the second order split operator method to propagate the initial wavepacket. Absorbing functions are introduced to avoid reflection at the end of the grids. Since we were focusing on the state-to-state at low collision energies, the optimization of the absorbing potential is crucial. Absorbing potential of the following form was applied in the calculations:

exp −∆

, ≤ ≤

= exp −∆ exp−∆ , ≤ ≤

(10)

where x indicates R or r. The first part of the absorbing potential was designed for absorbing the outgoing product with relative low translational energy. However, the second part was designed for damping the outgoing product with relative high translational energy. The state-to-state Smatrix elements are extracted by matching the asymptotic boundary condition using the reactant coordinate based (RCB) method [28-32]. Briefly, the “intermediate” coordinate method is adopted in the implementation of the RCB method [28, 29], where Rv is fixed at the projection plane Rv0, thus the product wave function has only two degrees of freedom, rv and θv。The fixed Rv combined with either (Rα, θα) or (rα, θα) provide a convenient platform for expressing the product ro-vibrational wave functions. In each time step, the overlap between the propagated



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wave packet and the product ro-vibrational wave functions are carried out on the intermediate either (Rα, θα) or (rα, θα) grids. Finally, the state-to-state ICS and DCS are then obtained [33]

σ jν ← j ν ( E ) = 0 0

dσ jν ← j0ν 0 (ϑ , E ) dΩ

π

1

∑∑ (2 J + 1) S JjνΩ← j ν

kν2j 2 j0 + 1 Ω ' Ω

J

0 0Ω

2 '

(E)

1 1 = (2 J + 1)d ΩJ ' Ω (ϑ ) S jJνΩ ← j ν Ω ' ∑ ∑ 0 0 2 j0 + 1 Ω ' Ω 2ik0 J

(11)

2

(12)

with k0 = 2µ R E and E is the collision energy. III. RESULTS AND DISCUSSIONS The relevant parameters in the TDWP calculations are listed below, which have been extensively tested for obtaining converged results. A total of 215 translational basis functions for the R coordinate were employed in the range of 0-22 a0 , the same number of translational basis functions for the r coordinate are used for 0.6-22 a0 , and jmax = 100 for rotational basis. The initial wave packet was propagated for a total propagation time of 180000 a.u., where the second order split operator with time step of 5 a.u. was applied for propagation. The initial wave packet covered collision energy in a range of 0 ~ 0.4 eV. Partial waves for total angular momentum J up to 40 are calculated for the CC calculations that the maximum K values are set to equal to min(J,

jmax ), which gives converged ICSs and DCSs only up to collision energy of 0.2 eV. We note that due to the long range potential of this ion-molecule reaction, the grid ranges for R and r applied in the calculations are quite long. The final product state-resolved information was extracted at Rv0 = 14.0 a.u.


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The titled ion-molecule reaction involves long range potential and this reaction proceeds without barrier. Thus, the quality of the absorbing is important since the reaction could happen at low collision energy. To illustrate the efficiency of our applied absorbing potential, the total reaction probabilities as a function of collision energies were presented in Figure 1, where the energy interval is small as 0.001 eV for a detailed justification. Absorbing potential with 4 sets of parameters is applied. In AP 1, xa = 18.7, xb = 22.7, xend = 23.0, Ca = 0.03 and Ca = 0.05 are applied. In AP 2, xa = 19.7, xb = 22.7, xend = 23.0, Ca = 0.015 and Ca = 0.05 are applied. In AP 3 and AP 4, xa = 19.7, xb = 22.7 and xend = 23.0 are kept as those in AP 2, however, the strength Ca was changed as 0.02 and 0.03. The total absorbing length for AP 1 is 4.3, and in the other three sets, the length is 3.3. For the numbers listed above, atomic units are adopted. From results in Figure 1, it is seen that the absorbing potential works very well for collision energy above 0.1 eV, since the total reaction probabilities hardly vary with different absorbing potentials. For collision energies below 0.02 eV, there is large uncertainty with the calculated results. It is not surprising since the absorbing potential only works well for the particles with wavelength much smaller than the length of the absorbing potential. Therefore, we believe that the results reported below for collision energies above 0.02 eV are credible. Figure 2 presents the total reaction probabilities as a function of collision energies with interval of 0.01 eV for several J values: (a) for J = 0, 5, 10, (b) for J = 15, 20, 25, and (c) for J = 30, 35, 40. The total reaction probabilities exhibit many resonance peaks, which are caused by the long-lived resonance states over the deep potential well on the ground surface of the titled reaction. As can be seen from Figure 2(a), with increasing values of collision energy, the reaction probabilities for all J values decrease slightly but do not change much. However, in Figure 2(b) and 2(c), the total reaction probabilities in this energy regime do not change for these several J



values and simply oscillate around a constant value. Anyway, the “constant” values decrease slightly with increasing value of J, which is seen clearly by a comparison between the results in Figure 2 (b) and (c). As shown in Figure 2, the “J-shifting” appears clearly in the J-dependent total reaction probabilities as a function of collision energies, caused by the centrifugal barrier. Figure 3 presents total reaction probabilities as a function of collision energies using the CS, CC and “truncated” CC approximation (where only limited number of K-components were adopted) for total angular momentum J = 30, for checking the validity of the K-truncation approximation for this reaction and thus any possibility of computational effort reduction. The Kmax, which was set in the calculation and defines the number of K-blocks, were chosen as 0 (CS approximation), 3, 10, and 20 in Figure 3 (a), (b), (c) and (d), respectively. It is seen that the effects of the Coriolis coupling are significant for this reaction. Comparing with the CS approximation, the threshold energy of the full CC results decreases much, which usually suggests the strong Coriolis couplings in a reaction. At the same time, the total reaction probabilities increase with increasing number of K-blocks. The agreement between the results with “truncated” CC approximation and those obtained by the rigorous CC calculations becomes better with increasing K-blocks, but detectable discrepancies persist if the K-blocks were not included completely in the calculations. Consequently, “truncated” CC approximation or CS approximation cannot give credible values of the ICS and DCS for this reaction thus full CC method have to be adopted for this reaction, as did in the following of the present work. This suggests that the CS approximation fails in describing the collision of the title system with a deep well which is consistent with the conclusion drawn in a review paper by Chu and Han (see Ref. [34] and the references in). In Ref. [7], statistical quantum method and the exact real Chebyshev wave packet method have been used by González-Lezana et al. to study the H+ + D2 and D+ + H2


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reactions, where they examined possible modifications of the usual CS approximation and various helicity decoupling schemes. They concluded that more K-blocks (which is Ω in their work) values need to be included for higher J for the H+ + D2 and D+ + H2 reactions. Our calculations are also consistent with their findings but for the D+ + HD reaction. The reaction probabilities as a function of the collision energies for different J values of the two product channels, D+ + HD'→D+ + HD' and D+ + HD'→H+ + DD', are presented in Figure 4. For HD channel, reaction probabilities do not change much with increasing collision energy, but simply oscillate around a constant value. They also do not change much with increasing total angular momentum J, except that the oscillations due to resonances reduce much with increasing J. For D2 channel, reaction probabilities decrease slightly as a function of collision energies. This is probably due to the fact that there is more chance to be bounced back for that the incoming D atom extracts the other D atom than that the incoming D atom brings away the H atom at high collision energies, caused by their difference in the masses of the H and D atom of the diatomic molecule. This fact could be understood by a picture in classical mechanics. During the reaction, the old bond of the reactant diatomic molecule need to be broken and new bonds between the D/H atom of the reactant diatomic molecule and the incoming D atom need to form, which require that the incoming D atom comes close enough to the D/H atom of the diatomic molecule. During the D atom incoming processes, the D atom possibly is bounced back by the D/H atom of the diatomic molecule or glances off. If the new bond forms, the incoming D atom then brings away the D/H atom of the diatomic molecule during the processes. In the case of failure to form the new bond, the incoming D atom is possibly bounced back or glances off too far from the diatomic molecule. Since the D atom in diatomic molecule has larger mass, which



correspondingly has larger inertia and is able to bounce back the incoming D atom with more chance at high collision energy. In contrast, the H atom makes faster response to the incoming of the D atom and to be brought away to form the product. The reaction probabilities of the D2 channel also decreases with increasing values of J. Since H atom has longer arm than D atom for the diatomic reactant molecule with respect to the diatomic centroid, with increasing J the D atom was screened by the H atom due to the rotation of the diatomic molecule, since the diatomic molecule acquires more rotational energy during the reaction with increasing J. Thus the incoming D atom has more chance to see H atom. In other words, the H atom of the reactant diatomic molecule has larger acceptance cone than the D atom. As a result, in the competing between the two channels, the titled reaction has more chance to proceed with product HD with increasing collision energy and total angular momentum J. The above pictures are drawn based upon a direct reaction process, which may be not exact for the titled reaction with potential well and need to be verified by a quasi-classical trajectory study, but would be helpful for qualitatively understanding the different behaviors of the two channels shown in Figure 4. From the plots in Figure 4, it is seen that the threshold for the reaction probabilities of the D2 channel shifts more rapidly to higher collision energy. This is because that the total reduced mass of the D2 channel is smaller than that of the HD channel, which leads to the fact that the reaction of the D2 channel feels larger centrifugal potential with the same value of J and more significant energy shifting for the profile of the reaction probabilities as a function of collision energy with increasing J.


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In Figure 5, opacity functions of the two channels at several typical collision energies are calculated by multiplying the total reaction probability with a factor of (2J + 1). With increasing collision energy, partial waves with J of larger values play more important roles in the reaction for both channels. Both of them at different collision energies first increase with increasing value of J, after reach their maximal values then decrease. However, the detailed variation trends of the opacity functions are different for the two channels. Due to larger centrifugal potential of the D2 channel, the opacity function drops rapidly with increasing values of J after reaching its maximum, e.g. in Figure 5(d) the probabilities of the D2 channel for J = 30 are much less than that of the HD channel in Figure 5(c) at higher collision energy. In contrast, the maximum of the opacity function for the D2 channel reached at smaller value of J than the HD channel. A comparison of ICSs of the two channels of the titled reaction is presented in Figure 6. Both ICSs decrease with increasing collision energy which are similar to the results of the isotopic reaction D+ + H2 in a previous study [34]. However, the ICSs in Figure 6 decrease much more rapidly than that of D+ + H2 due to the larger mass of HD in the D+ + HD in the present study, which may bounce back the incoming D atom more efficiently with increasing collision energy due to its larger inertia of HD molecule. We note here that the ICSs are only shown for collision energy higher than 0.02 eV, since the wavepacket method is difficult to deal with the reaction in the condition of low collision energies as we discussed above. The results in Figure 6 indicate that the ICS of the D2 channel are much larger than that of the HD channel for collision energy above Ec = 0.02 eV. It should be noted that with Ec = 0.01 eV, the ICSs of the HD channel are the dominant one. We do not show the results here is due to the poor convergence at those collision energies. Interestingly, the ICSs of both channels decrease with some oscillations with increasing collision energy, especially for the results of the



HD channel which exhibits a deep minimum around collision energy of 0.03 eV. The oscillations may result from the involved different intermediate states over the potential well where the nonadiabatic coupling is strong. These intermediate states control the product ratio for these two channels, i.e., one intermediate state may favor the product of HD but the other one may favor D2 product. The oscillations suggest the competitions between the two channels. In most collision energy regime, when the ICS decreases with increasing collision energy in one channel but increases in the other, giving rise to the valleys in the ICS of one channel but peak of the other. To investigate more details of the titled reaction, total DCSs as a function of collision energies of the two product channels are displayed in Figure 7 as contour plots. It is seen that for both channels, the product angular distribution is dominated by scattering in both forward and backward scatterings. However, the DCSs are not backward-forward symmetric and the asymmetry varies with collision energy. The DCSs of the HD channel presented in Figure 7(b) suggest that the product HD has more chance to be scattered into backward and forward direction than the D2 channel shown in Figure 7(a). For taking a closer examination of the DCS, total DCS at four different collision energies, 0.03, 0.08, 0.1 and 0.2 eV, for both channels are presented in Figure 8 and 9 as lines. It is seen that the angular distributions are quite different from each other for these two product channels. The DCS of the D2 channel which is presented in Figure 9(a) shows a significant forward scattering at collision energy of 0.03 eV. However, the DCS at collision energy of 0.03 eV is almost symmetrical for the HD channel in Figure 8(a). At other collision energies, the dominant scattering angles are almost opposite for these two product channels. Anyway, the DCSs do not exhibit rigorous forward-backward symmetry for all the DCSs shown in Figure 8 and 9. The


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oscillations in the figures fade out with increasing collision energy for both channels, and the profiles of the DCSs become smoother. Figure 10 displays the product ro-vibrational state-resolved ICSs for the D+ + HD' (v = 0, j = 0) → D'+ + HD (v' = 0, j') and D+ + HD' (v = 0, j = 0) → H+ + DD' (v' = 0, j') reactions at collision energy of 0.03 eV (a), 0.08 eV (b), 0.1 eV (c) and 0.2 eV (d). For the same collision energy, higher rotational states are produced in the D2 channel, e.g. j'max = 3 for the D2 channel but j'max = 1 for the HD channel for Ec = 0.03 eV, and j'max = 7 for the D2 channel but j'max = 5 for the HD channel for Ec = 0.2 eV, due to larger mass of the D2 molecule. On the other hand, the rovibrational state-resolved ICSs of product D2 are even larger than those of the product HD, which leads to larger total ICS for the D2 channel than that for the HD channel. This is especially true for Ec = 0.03 eV. Product ro-vibrational state-resolved DCSs are presented in Figure 11 and 12. With collision energies below 0.2 eV, product with vibrational excited states is not accessible. For the same product rotational state with increasing collision energy, the oscillations in the DCSs as a function of scattering angle are depressed due to more partial waves involved. The DCSs become smoother with increasing j' for the same collision energy. It is also seen that the profiles of the DCSs for different product rotational states appear quite randomly but they extend in the whole scattering angle, which is very typical for a reaction involving intermediate of long lifetime. All of the DCSs presented in Figure 11 and 12 do not exhibit rigorous backward-forward symmetry. This may indicate that even the titled reaction involving a deep potential well and many resonance states, the lifetimes of the intermediate resonance complexes should not be that long.



There have been quite some works on investigations of the product rotational state resolved DCS of H3+ system and its isotopic reactions, where the diatomic reactant exhibit symmetry thus they are computationally economic [see the review paper Ref.35 and the references in and Ref.36]. In Ref.[36], by using the time-independent quantum mechanical (TIQM) method, total and state-to-state DCSs at the collision energy of 0.001 and 0.005 eV for the D+ + H2 (v = 0, j = 0) → H+ + HD (v = 0, j') reaction were calculated. For j' = 1 and 2, the DCSs at 0.001 eV collision energy exhibit asymmetrical profiles in which the forward scattering dominates. However, DCS of the j' = 0 at 0.001 eV collision energy shows a nearly symmetric profile. All of the DCSs at collision energy Ec = 0.005 eV exhibit asymmetrical profiles, with the forward scattering is about twice of the amplitude of the backward scattering. In Ref.[36], the statistical quantum mechanical (SQM) method also were applied. All of the DCSs by the SQM show symmetric profiles at collision energies of 0.001 and 0.005 eV due to the random phase approximation, where the interference between different values of the total angular momentum J are neglected. Both the DCSs calculated by the TIQM and our wave packet method do not shown scattering with forward-backward symmetry, therefore, the SQM method may not works so well with the titled and its isotopic reactions for predicting the product state-resolved DCS. Anyway, the DCS of the titled reaction suggests a typical insertion reaction, which warrants the validity of the SQM for predicting most properties at other than the state-to-state level.

IV. CONCLUSIONS In summary, we performed an accurate quantum dynamics study of the titled reaction by a time-dependent wave packet approach on the diabatic KBNN potential energy surface. Total


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reaction probabilities, product ro-vibrational state-resolved ICSs and DCSs at a series of collision energies have been calculated for the two channels of the titled reaction. Due to the deep well in the potential energy surface of the reaction, rigorous CC calculation is required for the collision energy up to 0.4 eV. The titled reaction tends to be more reactive for the D2 channel whose ICSs are much more larger than that of the HD channel in the collision energy range from 0.02 eV to 0.2 eV. The DCSs do not show forward-backward symmetry and the asymmetry varies with product rotational states and collision energies for both channels. This may suggest that even the titled reaction involving deep potential well, which supports many resonance states, the lifetimes of the intermediate resonance complexes should not be that long, and the statistical theory works not so well for extracting detailed information about the titled reaction. A rigorous quantum reaction scattering theory has to be applied to predict more detailed properties, such as the differential cross sections, of the titled reaction.

FIGURES:

Figure 1. (Color online) Total reaction probabilities for the D+ + HD' (v = 0, j = 0) reaction as a function of collision energies with different absorbing potentials. Bottom panel are enlarged part of top panel for a closer reading. For parameters of the absorbing potential, please refer main text. The results for collision energies above 0.1 eV are not sensitive to the variation of the absorbing potentials. However, the results for collision energy below 0.02 eV may have large uncertainty.



Figure 2. (Color online) Total reaction probabilities for the D+ + HD' (v = 0, j = 0) reaction with collision energies up to 0.4 eV for several J values: (a) for J = 0, 5, 10, (b) for J = 15, 20, 25, and (c) for J = 30, 35, 40.

Figure 3. (Color online) Total reaction probabilities with J = 30 as a function of collision energies using the CC, CS and “truncated” CC approximation with Kmax blocks are set as 3, 10, and 20.

Figure 4. (Color online) Reaction probabilities for (a) D+ + HD'→D'+ + HD (channel 1) and (b) D+ + HD'→ H+ + DD' (channel 2) reactions with several typical J values.

Figure 5. (Color online) (a) Opacity functions for the HD channel at several typical collision energies. (b) The same as (a) but for the D2 channel.

Figure 6. (Color online) Total integral cross sections for the two channels as a function of collision energies.

Figure 7. (Color online) Differential cross sections with collision energies from 0.02 eV to 0.2 eV for the two reaction channels: (a) D+ + HD' → D'+ + HD and (b) D+ + HD'→ H+ + DD'.

Figure 8. Differential cross sections for the D+ + HD' → D'+ + HD reaction for several typical collision energies: (a) Ec = 0.03 eV, (b) Ec = 0.08 eV, (c) Ec = 0.1 eV and (d) Ec = 0.2 eV.

Figure 9. The same as Figure 8 but for D+ + HD'→ H+ + DD' reaction. Figure10. (Color online) Product rotational states distribution at several typical collision energies for the two reaction channels: (a) D+ + HD' → D'+ + HD and (b) D+ + HD'→ H+ + DD'.


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Figure 11. (Color online) State-to-state differential cross sections at several collision energies for the D+ + HD' (v = 0, j = 0) → D'+ + HD (v'= 0, j') reaction. The left panel for Ec = 0.03 eV and Ec = 0.08 eV but the right one for Ec = 0.1 eV and Ec = 0.2 eV.

Figure 12. (Color online) The same as Figure 11 but for the D+ + HD' (v = 0, j = 0) → H+ + DD' (v'= 0, j') reaction.

AUTHOR INFORMATION

Corresponding Author *Haixiang He Email: [email protected]

Present Addresses School of Chemistry & Chemical Engineering, Guangxi University, Nanning 530004, Guangxi, People’s Republic of China

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources National Natural Science Foundation of China and Natural Science Foundation of Guangxi province. ACKNOWLEDGMENT



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FIGURES:


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Figure 1:

Figure 2.



Figure 3:


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Figure 4:



Figure 5:


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Figure 6:



Figure 7:


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Figure 8:



Figure 9:


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Figure 10:



Figure 11:


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Figure 12:

A table of Contents Graphic:




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Accurate Quantum Wave Packet Study of the Deep ... - ACS Publications

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