MCTDH study on the reactive scattering of the Cl + HD reaction based on the neural-networks potential energy surface

Chemical Physics xxx (2017) xxx–xxx

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MCTDH study on the reactive scattering of the Cl + HD reaction based on the neural-networks potential energy surface q Qingyong Meng a,b a b

Department of Applied Chemistry, Northwestern Polytechnical University, Youyi West Road 127, 710072 Xi’an, China State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, 116023 Dalian, China

a r t i c l e

i n f o

Article history: Received 12 July 2017 In final form 9 November 2017 Available online xxxx Keywords: MCTDH Reaction resonance Reactive probability Cl + HD Vibration excited state

a b s t r a c t To study the dynamics resonances of the Cl + HD reaction which was proposed to proceed via abstraction mechanism with no clear resonances, we perform dynamics calculations by the multiconfiguration timedependent Hartree (MCTDH) method based on recently developed neural-networks potential energy surface (Science 347 (2015), 60). The HD molecule in v ¼ 0 (GS), v ¼ 1 (EX1), v ¼ 2 (EX2), and v ¼ 3 (EX3) states is used for the reactant. For GS, no distinctive resonance peak is found, while for EX1 two distinctive peaks at kinetics energies of 0:11 and 0:17 eV are investigated. These resonance peaks are well consistent with the previous results (Science 347 (2015), 60). Moreover, the present MCTDH calculations predict well-marked resonance peaks at 0:04; 0:05; 0:07, and 0:10 eV for EX2 and EX3, which indicates that anticipation of the chemical bond softening model (Science 327 (2010), 1501) is confirmed in this work. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction The chemical reaction occurs when reactants react each other to form products by rearranging the atoms at transition state (TS), which is the first-order saddle point of the potential energy surface (PES). However, the reactive complex can sometimes be weakly and transiently captured in the saddle-point region as metastable state [1–6]. Such so-called dynamical resonance profoundly influences chemical dynamics. Over the past decades, much attentions have been attracted for the dynamical resonance of the triatomic reactions, such as the F + H2/HD [2,7,8] and Cl + H2/HD [6,9,10] reactions. Since the Cl + HD reaction has recently received attention [6], we shall focus on its dynamical resonances by performing flux analysis of the multiconfiguration time-dependent Hartree (MCTDH) propagated wave function. The Cl + HD reaction which has been considered to proceed through direct abstraction mechanism, is an important benchmark in the study of chemical dynamics. Its direct abstraction mechanism makes it should proceed with no clear dynamical resonances when the reactant HD is in ground state but if HD is vibration excited there may exist dynamical resonances. In 2000, Kandel et al. [9] studied the Cl + HD (v ¼ 1; j ¼ 1; 2) by stimulated Raman pumping and resonance-enhanced multiphoton ionization detec-

q

Festschrift for the 70th Birthday of Prof. Dr. Hans-Dieter Meyer. E-mail address: [email protected]

tion of H and D resulting form the DCl and HCl formation, respectively. The experimental results were compared with quasiclassical trajectory (QCT) calculations on the G3 PES [11,12], but the agreement was rather poor. Also in 2000, Yang et al. [10] reported the time-dependent wave-packet propagation calculations for Cl + HD based on both G3 and BW2 PESs [11,12]. However, all of theoretical studies [9,10] did not find dynamical resonances until in 2015 Wang et al. [6] reporting the extremely short-lived reaction resonances in the Cl + HD (v ¼ 1) reaction investigated both experimentally and theoretically. They [6] also proposed that these resonances are supported by shallow wells on the vibration adiabatic potential caused by chemical bond softening in the TS region. In this work, we aim to obtain a deeply understanding the dynamical resonances of Cl + HD by performing MCTDH calculations on the basis of a recently developed PES [6] constructed through neural-networks (NN) approach, with special attention to those processes where the reactant HD are vibrationally excited. The NN PES [6] has been proved to be very accurate (the fitting error is roughly 1 meV) and suitable to dynamical calculations. The MCTDH method, on the other hand, was designed for the wave packet propagation dynamics of a polyatomic system. Comparing with the standard time-dependent wave packet propagation and time-independent quantum scattering methods, MCTDH has capable to accurately resolve the time-dependent Schrödinger equation consuming less computational cost.

https://doi.org/10.1016/j.chemphys.2017.11.004 0301-0104/Ó 2017 Elsevier B.V. All rights reserved.

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The rest of this paper is organized as follows; in Section 2, we will describe the theoretical framework, including the Hamiltonian and details of MCTDH calculations. Section 3 presents the numerical details, results, and discussions. Finally, Section 4 concludes with a summary.

0:75 Bohr 6 rd 6 8:0 Bohr; 1:75 Bohr 6 rv 6 8:0 Bohr, and 0 6 h 6 p, while 128; 125, and 64 grid points are used for rd ; rv , and h, respectively. Moreover, to ensure convergence in MCTDH and POTFIT calculations, the single-particle function (SPF) size of rd =rv =h ¼ 50=50=40 and single-particle potential (SPP) size of rd =rv =h ¼ 100=100=c (c means contracted mode) are used, respec-

2. Theoretical framework

tively. Using such SPP size, there are 104 terms in the potfitted PES. Moreover, we define the relevant grid points [16] as those with potential energy being less than 3.0 eV. On these relevant grid points, the root mean square (RMS) error of POTFIT is 0.039 meV, which is much smaller than the PES fitting RMS error (about 1 meV). Even on all grid points, the RMS error of POTFIT is only 0.1 meV that is still very small. These RMS errors clearly indicate the convergence of the POTFIT calculation. For clarity, the contour cut of the potfitted PES on the r d -r v plane with h ¼ p2 is shown in Fig. 2. The present MCTDH simulations are carried out for translational energies ranging from 0.0 to up to 0.6 eV (57.9 kJ/mol) while the HD molecules are initially in their first few vibration states with quantum number v ¼ 0; 1; 2; 3. We shall return to this topic later. The detecting probabilities, Pd ðEÞ and Pv ðEÞ, are respectively obtained by analyzing the flux of the wave packet fractions through the coordinates r d and rv . The reactive probability is then defined as P reac ðEÞ ¼ P v ðEÞ ¼ 1  Pd ðEÞ. At rd and rv , we put the complex absorbing potentials (CAPs) [26,27] with the form V CAP ¼ igðR  RCAP Þn , where the quantities n and g are the order and the strength of the CAP and RCAP marks the starting point of V CAP . Obviously, the CAP is set to zero for R < RCAP and V CAP for otherwise. To ensure that the wave packet do not leave CAP and return to the reactive center, the parameters n; g, and RCAP in the CAP are carefully determined by test calculations. Here we find

In this work, we use the Jacobi coordinates whose definitions are given in Fig. 1. If consider the triatomic reaction A + BC and let G be the center of mass, then from Fig. 1 one can define that rd is the distance between A and G, rv the bond length of BC, while h is the angle between r d and r v . Assuming that the total angular momentum of the system is zero, we can directly write the Hamiltonian operator as

H¼

1 @2 1 @2 1 1   þ 2 2M @r d 2l @r 2v 2Mr 2d 2lr 2v

þ Vðrd ; r v ; hÞ

!

1 @ @ sin h sin h @h @h



ð1Þ

where M ¼ 2:78 Dalton is the total mass of Cl + HD and l ¼ 0:67 Dalton is the reduced mass of the HD molecule and the potential energy function Vðr d ; r v ; hÞ was constructed by the NN approach [6]. Turning to the quantum dynamics calculations, since the MCTDH algorithms implemented in the Heidelberg MCTDH package [13], have been well reviewed elsewhere [14–22], we refer the reader to these references. In this work to ensure numerical convergence, we choose the fast Fourier transformation (FFT), exponential (EXP) function, and extended Legendre (LEG) function [23] as the primitive basis functions for the rd ; rv , and h coordinates, respectively. Starting with the nuclear wave function in MCTDH form [14–16], the working equations of motion (EOMs) [14,16,18–22] can be derived with the aid of Dirac-Frenkel variational principle. In the time-dependent case, after propagating the initial wave function j Wð0Þi, one obtains the time evolved wave function j WðtÞi, based on which further analyses, say reactive probability or time-dependent expectation value hWðtÞ j A j WðtÞi of an observable A, are possible. On the other hand, applying a time-independent variational principle to the wave function yields the EOMs for improved relaxation [24] and block improved (BLK) relaxation [25]. By improved relaxation and BLK calculations, a set of wave functions is relaxed to eigenstates of the Hamiltonian, and corresponding eigenvalues (energy levels) are obtained. 3. Results and discussions 3.1. Numerical details Now, let us give numerical details of the present work. The ranges of the coordinates (see Fig. 1) are set to be

Fig. 1. Definition of the Jacobi coordinates for the present triatomic reactive Cl + HD system where A ¼ Cl, B ¼ D, and C ¼ H. The G point represents the center-of-mass of the BC molecule, whose bond length is r v . Setting r d be the distance between A and G. The angle between HD and AG is h.

the v V rCAP

CAP

functions 2

of

d V CAP ¼ i  103  ðrd  6:0Þ3

r

and

3

¼ i  10  ðr v  6:5Þ . Obviously, one can use the relationship P v ðEÞ þ Pd ðEÞ ¼ 1 to inspect convergence of the MCTDH calculations and the flux analysis. Having constructed the whole Hamiltonian operator, which includes Eq. (1) and the CAP, the initial wave functions j Wð0Þi are constructed by MCTDH relaxation calculations. As shown above, the first few vibration states of the HD molecule are used ð0Þ

as reactants, while the Cl atom is located at rd ¼ 5:5 Bohr with an initial momentum of 10:0 au moving towards HD. Shown in Fig. 3 are reduced densities of these initial wave functions on the rd -r v plane with h ¼ p2 . Here we would like to emphasize that in computing the initial wave functions an artificial harmonic potenð0Þ 2

ð0Þ

tial 0:01  ðr d  r d Þ Hartree is used to fix Cl at rd ¼ 5:5 Bohr. Finally, the relaxed wave function is operated by the operator expðipz zÞ to assign a momentum of 10:0 au, where the minus sign means that Cl moves towards HD. 3.2. Time-dependent wave packet Having constructed the Hamiltonian operator H and the initial wave functions j Wð0Þi, wave packet propagation calculations are performed. For clarity, we denote the cases of HD in the v ¼ 0; v ¼ 1; v ¼ 2, and v ¼ 3 states to GS, EX1, EX2, and EX3, respectively. We show the reduced densities of these propagating wave packet in Fig. 4 (GS and EX1) and Supporting Information [28] (EX2 and EX3). Taking GS and EX1 as examples (see Fig. 4), one can find following three points. First, the initial momenta of both GS and EX1 cases are identical, making both wave packets entrance into the TS region synchronously when t P 35 fs. After reaction, at t P 95 fs each of wave packets splits into two parts. One part passes through the TS region and arrives at products, while the other part rebounds

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Fig. 2. The contour cuts on the r d -r v plane with h ¼ p2 of the PES for the present Cl + HD system. The definitions of r v and r d are given in Fig. 1. Here let the potential energy of the reactant group Cl + HD be zero.

to the reactant region. Moreover, we find that the wave packets of both GS and EX1 cases almost simultaneously leave the TS region. Indeed, this is not surprising that the same initial momenta lead to the similar time needed for passing through the TS region. But this point does not imply the same reactivity of HD in different vibration states, which will be discussed later in Section 3.3. Second, on the basis of the PES, one can optimize the TS geometry of r d ’ 2:3 Bohr, r v ’ 3:4 Bohr, and h ¼ p2 , which indicates that the potential barrier is very close to the diagonal line perpendicular to the reaction coordinate, that is, the barrier is not either an early barrier or a later barrier (see also Fig. 2). According to the Polanyi rules [29,30] for triatomic system, this implies that the vibration excited states of HD have capable to enhance the reactive probabilities, while the kinetic energy of Cl also has capable to influence the reactive probabilities. Third, from Fig. 3 it is not surprising that it is easy to find nodes in the wave packets of HD in vibration excited states and that there are one, two, and three nodes in the wave packet for HD in the v ¼ 1; v ¼ 2, and v ¼ 3, respectively. Moreover, from Fig. 4 (b) and Supporting Information [28], we can find that these nodes are kept during the reaction, which implies that HD in vibration excited states lead to the product in vibration excited states. This may be caused by the symmetric property of the barrier.

3.3. Reactive probability Given in Fig. 5 are the present reactive probability for the reaction (the left panel) and non-reaction (right panel) processes. In this section, unless other specified we shall mention potential energy with vibration energy correction. In Fig. 5 we set the potential energy of reactant Cl + H2 (denoted by U R ) as the original point of the kinetic-energy axis, that is set U R ¼ 0. For each of the four cases, summing over the probabilities of all possible processes, we obtain unit, which is consistent with the simple conservation law of matter. Moreover, we can also find following several points. First, given in Table 1 are the barrier heights with harmonic vibration energy corrections, the reaction thresholds, and the resonance peaks in the probability curves, which are estimated from Fig. 5. One can see from Table 1 the kinetic-energy thresholds of 0:150; 0:058; 0:011, and 0.001 eV for the GS, EX1, EX2, and EX3 cases, respectively. The threshold of GS (0.150 eV) is much smaller than the ab initio barrier of 0.24–0.26 eV for the Cl + HD reaction. This clearly indicates the quantum tunneling effect in this reaction. Moreover, the threshold of 0.06 eV for EX1 is very close to the previous experimental and theoretical value of 0.07 eV [6], which implies the accuracy of the present work.

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Fig. 3. The reduced densities of the 3D initial wave functions on the r d -r v plane with h ¼ p2 for (a) the ground state with v ¼ 0, (b) the first excited state with v ¼ 1, (c) the second excited state with v ¼ 2, and (d) the third excited state with v ¼ 3, where v is quantum number of the stretch vibration of the HD molecule. For clarity, we also show ð0Þ 2 the contour cuts on the same coordinate plane of the PES (see also Fig. 2). In computing the initial wave functions an artificial harmonic potential 0:01  ðr d  r d Þ Hartree is ð0Þ used to fix Cl at rd ¼ 5:5 Bohr.

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Fig. 4. The reduced densities of the 3D time-dependent wave functions on the r d -r v plane with h ¼ p2 when the (a) Cl + HD (v ¼ 0) and (b) Cl + HD (v ¼ 1) systems are going through the TS region (from t ¼ 25 fs to t ¼ 115 fs). Similar to Fig. 3 we also show the contour cuts of the PES. The similar figures for the system with v ¼ 2 and v ¼ 3 are shown in Supporting Information [28].

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Fig. 4 (continued)

Second, although the wave packet of each case passes though the TS region almost simultaneously, the reactive probabilities of these cases are very different. Clearly, at the same kinetic energy of Cl the

higher vibration excited state HD is in, the larger the reactive probability is, which implies the higher reactive potency of HD in higher vibration excited states. As discussed above, because of symmetrical

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Fig. 5. MCTDH energy-dependent reactive probabilities versus kinetics energy (in eV) for the Cl + HD ! HCl/DCl + D/H reaction (left panel) and Cl + HD ! Cl + HD (right panel). Here, the black, red, blue, or green lines represent the reactive probabilities where the Vibration quantum number, v, of HD is 0, 1, 2, and 3, respectively.

Table 1 The MCTDH results on the reactive probabilities of the Cl + HD (v ¼ 0; 1; 2; 3) reaction, comparing with the previous results [6] for the Cl + HD (v ¼ 1) ! H + DCl reaction. All of energy values are given in eV. The first column gives the vibration excited situations of HD in this work. The second, third, and fourth columns give the barrier heights with harmonic vibration energy corrections. The fifth and sixth columns gives the reaction thresholds estimated from the reactive probability curves. The other columns gives resonance peaks in the probability curves. Case

GS EX1 EX2 EX3 a b c d

Barriera

Threshold b

c

Cl + H2

Cl + HD

Cl + D2

MCTDH

0.242 — — —

0.261 (0.253) — — —

0.278 — — —

 0:150  0:058  0:011  0:001

Resonance d

TSWP

MCTDHc

TSWPd

—  0:070 — —

— 0.11, 0.17 0.05, 0.10 0.04, 0.07

— 0.10, 0.18 — —

The barrier heights are computed with vibration energy correction form PES. ‘‘Cl + HD” means both Cl + HD ! H + DCl and Cl + HD ! D + HCl. The barrier heights for the latter reaction are given in parentheses. The present work. Reference [6].

property of the TS (i.e. the property of non-early barrier and nonlater barrier) the vibration excited HD molecule can enhance the reaction. Third, there exist obvious dynamical resonances peaks in Fig. 5, in particular in the curves of the EX1, EX2, and EX3 cases. This resonance phenomena has been well discussed by Yang et al. [6] and reviewed by Guo [31]. By performing their time-dependent wave packet propagation (TSWP) method based on the NN PES for the Cl + HD (v ¼ 1) ! DCl + H reaction, Yang et al. [6] computed the reactive probability (the black solid line in Fig. 3B of reference [6]) and found the resonance peaks at collision energies of roughly 2:4 kcal/mol (0:10 eV) and 4:3 kcal/mol (0:18 eV). The resonance peaks are also reproduced in this work. For the EX1 case, the resonance peaks are found at  0:11 and  0:17 eV, which are very close to the previous results [6]. Here we would like to emphasize that in contrast to the previous study on Cl + HD (v ¼ 1) ! DCl + H, in the present work both Cl + HD ! DCl + H and Cl + HD ! HCl + D reactions are studied. This means that the present work must predict higher reactive probability than the previous one [6], and hence there cannot exist a superposition of the present probability and the previous one [6]. Finally, over the past decade to explain the dynamics resonances, Yang and collaborators [6–8] proposed the chemical bond softening caused by vibration adiabatic potential wells and predicted that the dynamical resonances exist in many reactions involving vibrationally excited molecules. Thus we further perform dynamics calculations for the EX2 and EX3 and give their reactive probabilities in Fig. 5 by blue and green lines, respectively. Form there it is easy to find resonance peaks which are more clear than

those of the EX1 case. For example, at the EX2 curve (blue line) there exist resonance peaks at roughly 0:05 eV and 0:10 eV, while at the EX3 curve (green line) two peaks at roughly 0:04 eV and 0:07 eV can be found. These results are clearly consistent with the prediction of the chemical bond softening model [6–8]. 4. Conclusions Since the Cl + HD reaction was proposed to proceed via a direct abstraction mechanism with no clear reaction resonances [9,10,6], MCTDH calculations based on the recently developed NN PES [6] are performed to study its dynamical resonances when the HD molecule is in the vibrational v ¼ 0 (GS), v ¼ 1 (EX1), v ¼ 2 (EX2), and v ¼ 3 (EX3) states. For the GS case, no distinctive resonance peak is found, which is agreement with the previous conclusion on the GS case [9,10,6]. For the EX1 case, two distinctive peaks at kinetics energies of 0:11 and 0:17 eV are investigated. These resonance peaks are very close to the previous resonance peaks of 0:10 and 0:18 eV, which implies the accuracy of this work. Furthermore, distinctive resonance peaks in the EX2 and EX3 cases are also found, which indicates that anticipation of the the chemical bond softening model is confirmed. Acknowledgments The financial supports by National Natural Science Foundation of China (Grant Nos. 21503214 and 21773186), Fundamental Research Funds for the Central Universities (Grant Nos. 3102017OQD035 and 3102017JC01001), and Hundred-Talent Pro-

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gram of Shaanxi are gratefully acknowledged. The author is grateful to Dr. Jun Chen (Xiamen University) for providing the subroutine of the PES and discussing the Cl + HD system. Parts of the calculations have been done on the supercomputing system in State Key Laboratory of Molecular Reaction Dynamics operated by Dalian Institute of Chemical Physics of Chinese Academy of Sciences. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.chemphys.2017.11. 004. References [1] R.D. Levine, S.-F. Wu, Chem. Phys. Lett. 11 (1971) 557–561. [2] G.C. Schatz, J.M. Bowman, A. Kuppermann, J. Chem. Phys. 58 (1973) 4023– 4025. [3] G.C. Schatz, Science 288 (2000) 1599–1600. [4] F. Fernandez-Alonso, R.N. Zare, Ann. Rev. Phys. Chem. 53 (2002) 67–99. [5] K. Liu, Adv. Chem. Phys. 149 (2012) 1–46. [6] T. Yang, J. Chen, L. Huang, T. Wang, C. Xiao, Z. Sun, D. Dai, X. Yang, D.H. Zhang, Science 347 (2015) 60–63. [7] M.H. Qiu, Z.F. Ren, L. Che, D.X. Dai, S.A. Harich, X.Y. Wang, X.M. Yang, C.X. Xu, D. Q. Xie, M. Gustafsson, R.T. Skodje, Z.G. Sun, D.H. Zhang, Science 311 (2006) 1440–1443. [8] W.R. Dong, C.L. Xiao, T. Wang, D.X. Dai, X.M. Yang, D.H. Zhang, Science 327 (2010) 1501–1502.

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