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Rotational and Isotopic Effects in the H + OH # H + HO+ Reaction Hongwei Song, Anyang Li, and Hua Guo J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b11574 • Publication Date (Web): 05 Jan 2016 Downloaded from http://pubs.acs.org on January 10, 2016

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The Journal of Physical Chemistry

Submitted to JPCA, 11/26/2015

Rotational and Isotopic Effects in the H2 + OH+ → H + H2O+ Reaction

Hongwei Song,& Anyang Li,# and Hua Guo* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico, 87131, USA

&

: Present address: State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, National Centre for Magnetic Resonance in Wuhan, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China # *

: Present address: College of Chemistry and Materials Science, Northwest University, Xi’an 710069, China : Corresponding author, email: [email protected]

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Abstract

Initial state selected time-dependent wave packet and quasi-classical trajectory methods are employed to study the effects of reactant rotational excitations and isotopic substitutions on the title reaction. The coupled-channel (CC) and/or centrifugal sudden (CS) integral cross sections are calculated quantum mechanically. It was found that the CS cross sections are slightly smaller than the CC counterparts over the collision energy range studied. The quantum dynamical and quasi-classical trajectory results agree reasonably well and both indicate that the rotational excitation of H2 enhances the reaction in all energies, while the rotational excitation of OH+ promotes the reaction more strongly at low collision energies but has a negligible effect at high collision energies. In addition, there exist significant isotopic substitution effects: The reaction cross section of the D2 + OH+ reaction is much lower than those of the H2 + OH+ and HD + OH+ reactions, which are quite close.

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I.

INTRODUCTION

Ion-molecular reactions have attracted considerable attention of both theoretical and experimental researchers in the last several decades owing to their important role in many fields, such as interstellar processes,1 electric discharges,2 and planetary ionospheres.3 Recent advances in experimental techniques have enabled detailed measurements of both integral4 and differential cross sections5 of some prototypical ion-molecule reactions, posing a challenge to accurate theoretical characterization of the reaction dynamics. This experiment-theory interplay has helped to provide a more in-depth understanding of issues pertaining to microscopic reaction mechanism and dynamics.6, 7 In this publication, we focus on the reaction between OH+ (X% 3Σ− ) and

H2 (X% 1 Σg+ ) , which leads to the formation of H( 2 S ) + H 2O+ ( X% 2 B1 ) . This is an important step in the interstellar formation of the HxO+ (x=1,2,3) species,8-10 which have recently been found in various interstellar media by the powerful Herschel space telescope.11-13 In addition to its astrochemical interest, this reaction also serves as a prototype in understanding reaction dynamics of ion-molecule reactions,14 The H2 + OH+ → H + H2O+ reaction mainly takes place adiabatically via the lowest triplet PES at low energies, of which the full-dimensional potential energy surface (PES) has recently been developed based on a permutation invariant polynomial-neural network (PIP-NN)15,

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fit of ~30,000 points calculated at the Davidson corrected 3



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multi-reference configuration interaction level with the aug-cc-pVTZ basis set.17 Particular attention has been paid to the long-range interactions between the two reactants to make sure the proper behavior. More recently, the thermal rate coefficients for the title reaction and its isotopologue have been computed on this PES at several temperatures and they agree with experimental values very well,14 underscoring the accuracy of the PES.

The reaction pathway on the ab initio PES for the title reaction, which is shown in Fig. 1a, bears much similarities to that of the reaction between H2O+ and H2.18, 19 Both reactions are exothermic and characterized by a submerged barrier and a shallow well in the product channel. Ng and coworkers have studied the H2O+ + H2 reaction and found surprising enhancements of the reaction by rotational excitations of the H2O+ reactant, particularly at low collision energies.18, 20, 21 This rotational effect was also found to result in a unique temperature dependence of the rate coefficient.22 The enhancement was attributed to coupling of the rotational mode of the H2O+ reactant with the reaction coordinate at the submerged saddle point,19 according to our recently proposed Sudden Vector Projection (SVP) model.23, 24 Interestingly, a SVP analysis for the title reaction also predicts a strong rotational effects for the OH+ reactant,17 which has been implicated in our recent combined experiment-theory study of the thermal rate coefficients using a quasi-classical trajectory (QCT) method.14 In this publication, we report the first initial state specific quantum dynamical (QD) calculations on the title reaction on the same PES, supported by additional QCT 4


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calculations. Such full-dimensional QD calculations have never been attempted for any ion-molecule reaction with more than three atoms, because of the huge computational costs needed to cover the long-range interaction between the reactants. This paper is organized as follows. Section II outlines the theoretical methodology. The results are presented and discussed in Section III. We conclude in Sec. IV.

II. THEORY

II A. Quantum dynamics

The time-dependent initial state-selected wave packet (ISSWP) method employed in this study is well documented in the literature.25 Here, we only briefly outline the key aspects. The full-dimensional Hamiltonian for the diatom-diatom reaction AB + CD in the reactant Jacobi coordinates, as shown in Fig. 1b, for a given total angular momentum J can be written as ( h = 1 hereafter):26 ˆj 2 ˆj 2 1 ∂2 ( Jˆ − ˆj12 ) 2 Hˆ = − + hˆ1 (r1 ) + hˆ2 (r2 ) + + 1 2+ 2 2 2 2 2 µ R ∂R 2µ R R 2 µ1r1 2 µ 2 r2 ref ref ˆ + V ( R, r , r ,θ ,θ ,ϕ ) − V (r ) − V (r ), 1

2

1

2

1

1

1

2

(1)

2

where R is the distance between the centers of mass (COMs) of AB and CD, r1 is the bond distance of AB, and r2 is the bond distance of CD. Their corresponding reduced masses are denoted as µR , µ1 , and µ2 , respectively. ˆj1 and ˆj 2 are the rotational angular momentum operators of AB and CD, respectively and they are coupled to ˆj12 .

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Jˆ

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is the total angular momentum operator of the system. hˆi (ri ) are the

one-dimensional (1D) reference Hamiltonians, which are defined as 1 ∂2 hˆi (ri ) = − + V ref (ri ), 2µ i ∂ri 2

(2)

i = 1,2

where V ref (ri ) is the corresponding 1D reference potential along the coordinate ri . The parity (ε) adapted wave function is expanded in terms of the body-fixed (BF) ro-vibrational basis functions: r r r

ψ JMε ( R, r1 , r2 ) =

∑F ν ε

n ,ν1 ,v2 j , K

JM n 1v2 jK

ε ˆ ˆ ˆ unv1 ( R)φν1 (r1 )φν 2 (r2 )Φ JM jK ( R, r1 , r2 ),

(3)

where n labels the translational basis functions, v1 and v2 are the basis indices for

r1 and r2 , j denotes ( j1 , j2 , J ) . The translational basis function, unv1 , is dependent ε on v1 due to the use of an L-shaped grid.27 Φ JM in Eq. (3) is the parity-adapted jK

coupled BF total angular momentum eigenfunction, which can be written as

ε Φ JM = (1 + δ K 0 ) −1 2 jK

2J + 1 J* j K [ DK ,M Y j j + ε (−1) j + j + j 8π 12

1

1 2

2

12 + J

D−JK* , M Y j jj − K ], 12

(4)

1 2

where DKJ , M is the Wigner rotation matrix.28 M is the projection of J on the j K space-fixed z axis, and K is its projection on the BF z axis that coincides with R. Yj112j2

is the eigenfunction of ˆj12 defined as Y j1j12j2K = ∑ j1ω j2 K − ω j12 K y j1ω (rˆ1 ) y j2 K −ω (rˆ2 ), ω

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(5)

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and yjm denotes the spherical harmonics. Note the restriction that ε (−1) j1 + j2 + j12 + J = 0 for K = 0 in Eq. (4). The centrifugal potential, i.e. ( Jˆ − ˆj12 )2 / 2µR R 2 in the Hamiltonian, which gives rise to the coupling between different K blocks in the BF representation, is given by

ε ε Φ JM ( Jˆ − ˆj12 ) 2 Φ JM = δ jj′{δ KK ′ [ J ( J + 1) + j12 ( j12 + 1) − 2 K 2 ] jK j ′K ′

(6)

−δ K +1K ′λJK+ λJK+ (1 + δ K 0 )1/ 2 −δ K −1K ′λJK− λJK− (1 + δ K 1 )1/ 2 } and the quantity λ is defined as λ±AB = [ A( A + 1) − B ( B + 1)]1 / 2 . Within the centrifugal sudden (CS) approximation,29,

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the coupling between

different K blocks is neglected, thus K becomes a good quantum number and is conserved. This approximation has been widely used in quantum scattering calculations and the results for activated bimolecular reactions have been reasonably accurate, but the accuracy deteriorates for barrierless complex-forming reactions.31 As a result, it is important to test its validity in different reactions.

The initial wave packet is constructed by the direct product of a localized Gaussian wave packet in the scattering coordinate and a specific ( JK ε ) state of the reactive system with specific ro-vibrational states of both H2 and OH+ and the form and parameters are given in Table I. The wave packet is propagated in time using a second order split-operator method.32 The flux through the dividing surface,

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S[r1 = r1F ] , is obtained from the energy-dependent scattering wavefunction, ψ i+ ( E ) , which is determined by Fourier transforming the time-dependent wave packet at the dividing surface, placed in the product arrangement channel. In each step, the wave packet is transformed between different representations that diagonalized various terms in the Hamiltonian.33 Finally, the wave packet is absorbed at the edges of the grid using a negative imaginary absorbing potential with the form given in Table I in order to enforce outgoing boundary conditions.

The integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all the partial waves labeled by J,

σv v

1 2 j1 j2

1 (2 j1 + 1)(2 j2 + 1) 1 = (2 j1 + 1)(2 j2 + 1)

(E ) =

π

∑ε k ∑ (2 J + 1) P ε 2

j12 K

∑σ

j12 K ε

J v1v2 j1 j2 j12 K

J ≥K

j12 K ε v1v2 j1 j2

(E)

(7)

( E ),

j Kε where σ v112v2 j1 j 2 is defined as the j12, K and ԑ specific cross section and K is taken from

0 to min(j12, J). In this study, we focus on the dynamics from the ground vibrational states, i.e. v1 = v2 = 0 . Thus, we will drop the vibrational quantum numbers hereafter. The numerical parameters employed in the QD calculations of H2/HD/D2 + OH+ on an L-shaped grid are given in Table I. The parameters, which were well tested to give converged results, result in a very large size for the wave packet. The total number of grid points and basis functions for our calculations is approximately 1.5× 108 in the CS calculation and 6.0×108 in the CC calculation, underscoring the 8


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difficulties associated with such calculations. The propagation time is around 40000 a.u. with a time step of 10 a.u.

II B. Quasi-classical Trajectory Method

Standard QCT calculations for the title reaction were carried out based on the PES described above, using VENUS.34 The total reactive ICS for the reaction was computed according to the following formula: 2 σ r ( Ec ) = πbmax ( Ec ) Pr ( Ec ) ,

(8)

where the reaction probability at the specified collision energy Ec is given by the ratio between Nr and Nt:

Pr ( Ec ) = N r / N t .

(9)

The reactive differential cross section (DCS) is computed by dσ r σ P (θ ) , = r r dΩ 2π sin( θ )

(10)

where Pr( θ ) is the normalized probability for the scattering products at the scattering angle θ , which is given by

 υvi ⋅υv f θ = cos  v v  υi υ f  −1

 .  

(11)

v Here, υ is the relative velocity vector, and the subscripts ‘i’ and ‘f’ denote ‘initial’

v v v v v v and ‘final’, respectively, υi = υH 2 − υOH+ , υ f = υ H − υ H

2O

+

. The signs of the relative

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velocity vectors were chosen such that θ =0° corresponds to forward scattering and θ =180° corresponds to backward scattering.

The trajectories were initiated with a reactant separation of 10.0 Å, and terminated when products reached a separation of 7.0 Å, or when reactants are separated by 9.0 Å for non-reactive trajectories. During the propagation, the gradient of the PES was obtained numerically by a central-difference algorithm. The propagation time step was selected to be 0.02 fs, which conserves the energy better than 0.04 kcal/mol for most trajectories. Batches of 50 000 trajectories have been run at each collision energies (from 0.05 to 1.00 eV). A few trajectories which failed to converge energy to 0.04 kcal/mol or were nonreactive after 4.0 ps were discarded. The maximal impact parameter (bmax) was determined using small batches of trajectories with trial values.

III.

RESULTS

III A. Accuracy of the CS approximation

The CS approximation greatly reduces the computational costs and thus is widely employed in reaction dynamics studies of bimolecular reactions. Under the CS approximation, only one K block (i.e. the initial K) is included in the QD calculation. Here, we will test the accuracy of the CS approximation for the title reaction by comparing its cross section with the coupled-channel (CC) result. In the CC

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calculations, the value of K is taken from 0 to min(J, 2). Tests showed that larger Kmax values do not change the results significantly.

The CC and CS ICSs for the reaction with the ground ro-vibrational states of both reactants are presented in Fig. 2. Both are monotonically decaying functions with the collision energy, consistent with the barrierless nature of the reaction. The small kinks at low collision energy are presumably due to resonances supported by the well in the entrance channel (see Fig. 1a). From the figure, it is clear that the CS cross section is slightly but noticeably smaller than the CC counterpart over the entire collision energy range. The relative error is less than 14% at collision energies larger than 0.2 eV while it runs up to 25% at low collision energies. The errors, which are presumably caused by the fact that the shallow well in the entrance valley (see Fig. 1a) strengthens the Coriolis coupling among the helicity channels, are acceptable. Given the significant savings associated with the CS approximation, we report below the results obtained with this approximation.

III B. Rotational effects of H2 and OH+ Figure 3(a) shows the QD and QCT ICSs for the H2(j1 = 0-2) + OH+(j2 = 0) reaction as a function of total energy, which is the sum of collision and internal energies of the reactants. The QD excitation functions for the three initial rotational states of H2 are very similar, i.e. they first decrease sharply at low energies and then become more or less flat at higher energies. On the other hand, the QD cross section 11



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increases with the H2 rotational excitation over the entire energy range studied. In other words, low-lying rotational excitations of H2 clearly enhance the reactivity of this reaction. These features are well reproduced by the QCT results. However, it appears that the QCT cross sections are slightly larger than the QD counterparts at low energies. The QD and QCT ICSs for the H2(j1 = 0) + OH+(j2 = 0-2) reaction are plotted in Fig. 3b as a function of total energy. Clearly, both the QD and QCT results indicate that the rotational excitation of OH+ significantly promotes the reaction at low energies. At higher energies, rotational excitations of OH+ have a negligible effect on the reaction and the cross sections from the three rotational states of OH+ are nearly indistinguishable. This feature is also found in the H2 + H2O+ reaction, in which the enhancement effect is also much more pronounced at low collision energies and becomes weaker at high collision energies.18, 20, 21 This trend is reproduced by QCT results shown in the same figure. It should be noted that the enhancements of the title reaction by OH+ rotational excitations at low energies are much larger than that by H2 rotational excitations, particularly considering the rotational constant of OH+ (16.79 cm-1) is significantly smaller than that of H2 (60.85 cm-1). In order to better understand the rotational enhancement effects observed here, in Fig. 4, the quantum opacity function for j1=j2=0 is plotted as a function of the total angular momentum quantum number with the translational energies at 0.05, 0.2, and 12


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0.5 eV. The opacity functions first increase gently with the collision energy, reach maximum, and then decrease sharply. Clearly, the cross section is dominated by the partial waves near the highest J values allowed at particular energy. Now let us examine the reaction path for the title reaction at different J values. In Fig. 5, the sum of the interaction potential and centrifugal potential ( J ( J + 1) / 2µ R R 2 ) along the minimum energy path is shown as a function of the reaction coordinate. It is clear from the figure that the effective barrier changes with the J value.

At low J values (

Rotational and Isotopic Effects in the H2 + OH+ ... - ACS Publications

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