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Volume 35 Issue 2
Apr.  2022
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Article Contents
Abstract
Ⅰ.   INTRODUCTION
Ⅱ.   PES AND TDWP THEORETICAL METHODS
Ⅲ.   RESULTS AND DISCUSSION
Ⅳ.   CONCLUSION
Ⅴ.   ACKNOWLEDGMENTS
References
Related Articles
Hong-yu Ma, Cheng-yuan Zhang, Yu-zhi Song, Feng-cai Ma, Yong-qing Li. Time-Dependent Quantum Wave Packet Calculation for Reaction S−(2P)+ H2(1Σ+g)→ SH−(1Σ)+H(2S) on Ab Initio Potential Energy Surface[J]. Chinese Journal of Chemical Physics , 2022, 35(2): 338-344. DOI: 10.1063/1674-0068/cjcp2111239
Citation: Hong-yu Ma, Cheng-yuan Zhang, Yu-zhi Song, Feng-cai Ma, Yong-qing Li. Time-Dependent Quantum Wave Packet Calculation for Reaction S−(2P)+ H2(1Σ+g)→ SH−(1Σ)+H(2S) on Ab Initio Potential Energy Surface[J]. Chinese Journal of Chemical Physics , 2022, 35(2): 338-344. DOI: 10.1063/1674-0068/cjcp2111239
shu

Time-Dependent Quantum Wave Packet Calculation for Reaction S−(2P)+ H2(1Σ+g)→ SH−(1Σ)+H(2S) on Ab Initio Potential Energy Surface

  • a.

    Department of Physics, Liaoning University, Shenyang 110036, China

  • b.

    Collaborative Innovation Center of Light Manipulations and Applications, Shandong Normal University, Jinan 250358, China

More Information
  • Corresponding author:

    Feng-cai Ma, E-mail: fcma@lnu.edu.cn

    Yong-qing Li, E-mail: yqli@lnu.edu.cn

  • Received Date: November 18, 2021
  • Accepted Date: January 25, 2022
  • Available Online: March 02, 2022
  • Issue Publish Date: April 26, 2022
    Graphical Abstract
    • Abstract
  • The time-dependent wave packet propagation method was applied to investigate the dynamic behaviours of the reaction S−(2P)+H2(1Σ+g)→ SH−(1Σ)+H(2S) based on the electronic ground state (2A′) potential energy surface of the SH2− ionic molecule. The collision energy dependent reaction probabilities and integral cross sections are obtained. The numerical results suggest that there are significant oscillation structures over all the studied range of the collision energies. The vibrational excitation and rotational excitation of the diatomic reagent H2 promote the reactivity significantly as suggested by the numerical total reaction probabilities with the initial rotational quantum number of j=0, 2, 4, 6, 8, 10, and the vibrational quantum number v=0, 1, 2, 3, 4. The numerical integral cross sections are quite consistent with the experimental data reported in previous work.
    • FullText(HTML)

  • Sulfur element is one of the most abundant elements in the Earth's crust, and generally found as a free form near craters or in rocks in the Earth's crust. The physicochemical reactions involving elements containing sulfur species have aroused great attention due to their important role in combustion and atmosphere chemistry. Chemical reaction S−+H2, as an elementary chemical reaction, has several possible product channels [1], such as exothermic associative detachment S−+H2→SH2+e−, collisional detachment S−+H2→S+H2+e−, reactive charge transfer S−+H2→SH+H−, and reactive detachment S−+H2→SH+H+e− [2-6]. In 2000, Ervin et al. [1] investigated the transfer reaction of H atom within a certain range of energy, where associations are detached. However, the channel of electron and H-product can not be detected from the experiments, when the reaction was below the threshold. Employing guided ion beam tandem mass spectrometry, Rempala et al. researched the total reaction cross sections of the endoergic reactions between S− and H2 molecules experimentally, and put forward the hypothesis that neither rotational energy nor spin-orbital energy can accelerate the response close to the threshold. According to the electronic state correlation and characteristics of potential energy surface (PES), the reaction threshold of S−(2P)+H2(1Σ+g)→SH−(1Σ)+H(2S) was 14.10 kcal/mol. With the development of molecular dynamics [7-13], it is extremely important to obtain an accurate PES of polyatomic molecule, and understand the dynamic behaviors including reaction cross sections using quantum mechanical (QM) method.

    The first global PES of SH2− (2A′) was researched in 2018 [14]. More than 3000 ab initio points have been computed to determine all the forms of analytic function. Employing aug-cc-pVTZ(AVTZ) and aug-cc-pVQZ(AVQZ) basis sets with extrapolation of the electron correlation energy to the complete basis set (CBS) limit, all the accurate energies were calculated. The PES of the electronic ground state (2A′) of S−+H2 has a deep potential well (14.31 kcal/mol) relative to the asymptote of S−+H+H, and it fits pretty well with previous result [1]. To vertify the accuracy of the new PES, in this work, we will calculate the parameters of reaction S−+H2 and discuss the dynamic information with a time-dependent wave packet (TDWP) approach reported by Léforestier [15], and developed by Gray, Gomez-Carrasco and Zanchet et al. [16-23]. In the calculation of TDWP, the time-dependent Schrödinger equation (TDSE) can be solved using the split operator propagator methods [24]. Then the total reaction probability will be achieved after obtaining the dividing surface reaction flux. Recently, Sun et al. put forth a reactant coordinate based wave-packet method to extract product state-resolved attributes in both product channels [25]. Lin et al. discussed the implementation of an efficient and accurate wave-packet method for calculating S−matrix elements and thus DCSs for atom-diatom reactive scattering [26]. TDWP method has been successfully applied to investigate atom-diatom reaction dynamics, such as F+HD [27], He+H2 [28], H++D2 [29], S(1D)+HD [30], H+HS [31], He+H2+ [32], and S+D2 [33], etc. Based on the new PES, we calculate the reaction probabilities and integral cross sections (ICSs) of S−+H2 and then discuss in detail how the vibrations and rotations of the reactants contribute to the reaction, in order to provide valuable insights for understanding some related chemical reactions.

    A   PES

    SH2−(2A′) PES was calibrated in previous work [14] from ab initio points counted at the MRCI [34] level employing the full valence complete active space (FVCAS) [35] reference function, based on AV(T, Q)Z basis sets of Dunning [36, 37] and uniform single-pair and triple-pair (USTE) [38, 39] method by extrapolating the total energy to the CBS limit. Among the theoretical studies of PES, we find several major stationary points, including one global minimal (GM), two transition states (TSs) and three saddle points (SPs). The configuration of GM is found to be R1=3.654 a0, R2=R3=2.534 a0, θ=92.3∘ (R1 is the HH interatomic distance, R2 and R3 are the interatomic distances of SH−) and the dissociation energy is 0.197 Eh. While in TS1, R1 is equal to 2.47 a0, R2=2.63 a0, R3=4.5 a0, the bond angle θ is 27∘. TS2 has a configuration of R1=5.75 a0, R2=R3=2.87 a0, and the bond angle is linear. The topographical features of the new SH2−(2A′) PES were examined in detail, showing a good agreement with theoretical and experimental results. FIG. 1 presents a T shape contour plot of S− insertion into H2 with the fixed angle, by changing the bond lengths of x and y. The minimum reaction path (MEP) of S−(2P)+H2(1Σ+g)→SH−(1Σ)+H(2S) as a function of RH2−RSH− at fixed angle θ=90∘ is provided in FIG. 2. According to our theoretical results from this figure, the reaction is exdothermic with energy of 0.0913 eV.

    Figure  1.  The contour plot for the C2v symmetry of S− insertion into H2.
    DownLoad: Full-Size Img PowerPoint
    Figure  2.  The minimum reaction path of S−(2P)+ H2(1Σ+g)→SH−(1Σ)+H(2S).
    DownLoad: Full-Size Img PowerPoint

    B   TDWP method

    Due to the accuracy of PES directly determining the accuracy of subsequent dynamics calculation results, it is necessary to discuss the dynamics behaviours of the title system to test the accuracy of the new PES reported in Ref.[14]. The TDWP is an effective approach for investigation of reactive scattering and has been extensively described in previous works [22, 28, 40]. The main purpose of using it is to solve the TDSE of the reaction system on a grid (R, r, θ) employed in Body-fixed (BF) frame.

    In Jacobi coordinates, the Hamiltonian of SH2−(2A′) reaction should be represented

    H=−h22μR∂2∂R2+(J−j)22μRR2+j22μrr2+V(R,r,θ)+h(r)
    		 (1)

    where R represents the distance from S− to molecular center of mass of hydrogen, while r represents the distance between hydrogen molecules. θ is the angle between R and r. μR and μr correspond to the reduced masses, J is the total angular momentum and j is the rotational angular momentum of H2 molecule, V(R,r,θ) is the interaction potential of the title system, which is obtained from an accurate global PES [14], and h(r) is the diatomic reference Hamilton.

    In adiabatic approximation, the wave function just describes the motion of the nucleus. In general, the wave function of the TDSE [22, 41, 42] can be expanded as

    ψJKεv0j0k0(ˆR,ˆr,t)=∑nvjkFJKenvjk,v0j0k0(t)uvn(R)⋅ϕv(r)YJKεjk(ˆR,ˆr)
    		 (2)

    where K and k are the projection quantum numbers of the J on the space-fixed z-axis and BF z-axis, respectively. v0, j0, k0 represent the initial vibrational state, ε is the parity of the system and uvn(R) is the normalized wave function base vector [22, 42],

    uvn(R)=√2RLsinnπRRLRL=Rmax−Rmin
    		 (3)

    In Eq.(2), ϕv(r) is the basis vector of diatomic vibration wave function, and the satisfied equation is shown below

    h(r)ϕv(r)=εvϕv(r)
    		 (4)

    YJKεjk(ˆR,ˆr) is an eigen-function of (J2,j2,JZ). To save computing time, it is generally expressed as Ref.[22, 42]:

    YJKεjk(ˆR,ˆr)=(1+δk0)−1/2√2J+18π[DJk,K+ε(−1)J+KDJ−k,K]Yjk(θ,0)
    		 (5)

    The initial wave packet is [22, 41, 42]

    Ψ(0)=φk0(R)ϕv0,j0(r)YJKεj0k0(ˆR,ˆr)
    		 (6)

    where the wave packet φk0(R) is taken as the normalized Gaussian function,

    φk0(R)=(1πδ2)1/4exp[−(R−R0)22δ2]e−ik0R
    		 (7)

    δ is the width of the wave packet,

    E(k0)=12μR(k02+12δ2)
    		 (8)

    ϕv0,j0(r) for the diatom is also expanded in the vibrational basis function ϕv(r).

    In the adiabatic process [43], we will adopt the split operator method to propagate the wave function, which is expressed as follows,

    Ψ(t+Δ)=e−iH0Δ/2e−iVrotΔ/2e−iVΔ⋅e−iVrotΔ/2e−iH0Δ/2Ψ(t)
    		 (9)

    for a time step Δ, the wave function at time t+Δ is given by the wave function at time t:

    H0=−h22μR∂2∂R2−h22μr∂2∂r2+Vr(r)
    		 (10)
    Vrot=(ˆJ−j2)2μRR2+ˆj22μrr2
    		 (11)

    The reaction probability is given by [41],

    PJj0k0v0(E)=hmsIm[⟨Ψ+(E)|δ(s−s0)∂∂s|Ψ+(E)⟩]
    		 (12)

    The formula Ψ+(E) is a time independent final state wave function,

    |Ψ+(E)⟩=1a(E)∫∞−∞eiEt|Ψ(t)⟩dt
    		 (13)
    a(E)=⟨Ψ+(E)|Ψ(0)⟩=⟨ΦE|Ψ(0)⟩
    		 (14)

    By summing all the final states of the probabilities, the ICSs [41] are obtained. The formula of ICS is

    σj0v0(E)=πk2j0v0∑J(2J+1)PJj0v0(E)
    		 (15)
    PJj0v0(E)=12j0+1∑k0PJj0k0v0(E)
    		 (16)

    In the calculation of TDWP, we take the range of R in the Jacobi coordinates from 0.1−20 a0 and 250 translational basis functions, the range of r is 0.5− 17 a0 and 110 vibration basis functions. For obtaining converged ICS over the studied collision energy range, we take Jmax=80, the total propagation time up to 90000 a.u., and the scattering coordinate of the center of the initial wave packet is set as 15 a.u. with average translation energy 1.0 eV and width 0.27 a.u. The selection of these parameters are listed in Table Ⅰ, ensuring the convergence of the calculation results.

    Table  Ⅰ.  The parameters used in TDWP calculation.
     | Show Table
    DownLoad: CSV

    FIG. 3 shows, the reaction probabilities of the reactant H2 as a function of collision energy with J=0, 10, 20, 30, 40 in the ground vibrational state. As seen from this figure, each of all reaction probabilities presents a dense oscillation structure in the whole range because of the existence of a potential well in the reaction. Moreover, the vibration is intensely favored by a lower total angular momentum. The smaller the J, the more obvious the oscillation, especially for the case of J=0. The threshold energy, however, presents a movement to the higher energy with the increasing of J. The response threshold for different angular momentum in this figure are consistent with the characteristics shown in endothermic reaction.

    Figure  3.  The reaction probability of S−+H2 reaction for J=0, 10, 20, 30, 40.
    DownLoad: Full-Size Img PowerPoint

    To ensure the convergence of calculating reactant H2 under ground state vibration, the J-dependent partial wave contributions to the ICS are shown in FIG. 4 with the energy from 0.6 eV to 1.2 eV. As the collision energy increases, the distribution of the J value shifts to a higher peak. At 1.2 eV of the collision energy, a partial wave contribution is required to reach a total angular momentum of 55.

    Figure  4.  Weighted partial wave contributions to the ICSs.
    DownLoad: Full-Size Img PowerPoint

    The ICS of S−+H2 is shown in FIG. 5 for the collision energy over the range of 0.5 eV to 1.5 eV. Obviously, the ICS increases drastically with the increasing of collision energy, which is basically consistent with the experimental results reported in Ref.[1]. In addition, ICS has a threshold because of the high barrier in MEP. The trajectory of this curve is very alike to that for endothermic reactions. Comparing with FIG. 2, we can conclude that our kinetic results are in good agreement with those predicted by the MEP of PES. The influence of improving the vibrational energy of H2 on the reaction probability is presented in FIG. 6. As the collision energy increases, the reaction probabilities also increase. The reaction probabilities of high vibration quantum numbers are higher than ones with the low vibration quantum numbers. It indicates that, at low energy collisions, the H2 vibration increases the probability of the reaction. However, the strong oscillation characteristics of ICS is obvious for v=0 and there is a non-zero threshold compared with the others.

    Figure  5.  Comparison of ICS computed in this work with the one using guided ion beam tandem mass spectrometry by Katarzyna Rempala et al. [1] for S−+H2 reaction.
    DownLoad: Full-Size Img PowerPoint
    Figure  6.  The reaction probabilities of S−(2P)+H2(1Σ+g) (J=0) for different vibration quantum numbers v.
    DownLoad: Full-Size Img PowerPoint

    To understand the effects of rotation, we describe the probabilities of the S−+H2(J=0) reaction for j=0, 2, 4, 6, 8, 10 in FIG. 7, and the probabilities of the S−+H2(J=10) reaction in FIG. 8, respectively. Comparing the two figures, the increase in total angular momentum attenuates the oscillation in the probability diagram. For every initial rotational level, the reaction probability increases with the increasing of collision energy. In the case of low-energy collision, the rotation of H2 will promote the reaction. With the collision energy increasing, the rotation has no obvious promoting effect on the reaction. It can be concluded that rotation of H2 at low energies promotes endothermic and essentially obstructive reactions of S− atoms near H2 molecules. Meanwhile, in FIG. 9 we calculate the rate constant of S−+H2 reaction, which can provide some effective information for future researchers in solving experimental results.

    Figure  7.  The reaction probabilities of S−+H2(J=0) for the different rotation quantum number j.
    DownLoad: Full-Size Img PowerPoint
    Figure  8.  The reaction probabilities of S−+H2(J=10) for the different rotation quantum number j.
    DownLoad: Full-Size Img PowerPoint
    Figure  9.  The rate constant of S−+H2 reaction calculated in this work.
    DownLoad: Full-Size Img PowerPoint

    Based on the new globally accurate PES for ground state (2A′) of SH2−, the dynamical calculations for the reactions S−(2P)+H2(1Σ+g)→SH−(1Σ)+H(2S) in the collision energy range from 0.5 eV to 1.5 eV are investigated using a TDWP method. As the collision energy range varies from 0.5 eV to 1.5 eV, the obtained results indicate that the reaction probability presents a oscillation structure, because of the existence of potential well in the reaction. As the total angular momentum J increases, however, the amplitude of oscillations is weakened. The behavior of the v-dependent and the j-dependent reaction probabilities show that the vibrational excitation of reactant H2 and the rotation promote the reactivity throughout the collision energy range. The energy dependence of ICS presents a rising trend, which is in good agreement with the characteristic of endothermic reaction and available experimental data, indicating that our TDWP calculation are accurate and reasonable.

    This work was supported by LiaoNing Revitalization Talents Program (No.XLYC2007094), the Liaoning BaiQianWan Talents Program, the Natural Science Foundation of Liaoning Province (No.2020-BS-083), and the National Natural Science Foundation of China (No.11874241).

    †Part of Special Issue “In Memory of Prof. Nanquan Lou on the occasion of his 100th anniversary”.

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