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Time-Dependent Quantum Wave Packet Study of the Si + OH # SiO + H Reaction: Cross Sections and Rate Constants Alejandro Rivero Santamaría, Fabrice Dayou, Jesus Rubayo-Soneira, and Maurice Monnerville J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b00174 • Publication Date (Web): 07 Feb 2017 Downloaded from http://pubs.acs.org on February 11, 2017

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The Journal of Physical Chemistry

Time-Dependent Quantum Wave Packet Study of the Si + OH → SiO + H Reaction: Cross Sections and Rate Constants Alejandro Rivero Santamar´ıa,† Fabrice Dayou,‡ Jesus Rubayo-Soneira,¶ and Maurice Monnerville∗,† †Laboratoire de Physique des Lasers, Atomes et Molécules, UMR 8523 du CNRS, Centre ´ d’Etudes et de Recherches Lasers et Applications, Université Lille I, Bât. P5, 59655 Villeneuve d’Ascq Cedex, France ‡LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, F-75252, Paris, France ¶Departamento de F´ısica General, Instituto Superior de Tecnolog´ıas y Ciencias Aplicadas, Habana 10600, Cuba E-mail: [email protected]

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Abstract The dynamics of the Si(3 P)+OH(X2 Π) → SiO(X1 Σ+ )+H(2 S) reaction is investigated by means of the time-dependent wave packet (TDWP) approach using an ab initio potential energy surface recently developed by Dayou et al. (J. Chem. Phys. 2013, 139, 204305) for the ground X 2 A′ electronic state. Total reaction probabilities have been calculated for the first fifteen rotational states j = 0 − 14 of OH(v = 0, j) at a total angular momentum J = 0 up to a collision energy of 1 eV. Integral cross-sections and state-selected rate constants for the temperature range 10-500 K were obtained within the J-shifting approximation. The reaction probabilities display highly oscillatory structures indicating the contribution of long-lived quasibound states supported by the deep SiOH/HSiO wells. The cross sections behave with collision energies as expected for a barrierless reaction and are slightly sensitive to the initial rotational excitation of OH. The thermal rate constants show a marked temperature dependence below 200 K with a maximum value around 15 K. The TDWP results globally agree with the results of earlier quasi-classical trajectory (QCT) calculations carried out by Rivero-Santamaria et al. (Chem. Phys. Lett. 2014, 610-611, 335-340) with the same potential energy surface. In particular, the thermal rate constants display a similar temperature dependence, with TDWP values smaller than the QCT ones over the whole temperature range.

1 Introduction The SiO molecule is the most widespread silicon-bearing molecule in the interstellar medium 1 (ISM). It is observed in a variety of astrophysical environments, with abundances greatly dependent of the physical conditions of the media. While SiO is almost absent in quiescent cold clouds, its abudance increases significantly in warm shocked layers of outflows associated with nascent stars. The SiO molecule is thus considered as an excellent tracer of shocks in regions of star formation. 2–5 According to astrophysical models, 4,6–8 the Si+OH reaction is expected to be one of the major formation pathways of interstellar SiO in the gas-phase. Another possible pathway is the Si+O2 reaction, which has been studied both experimentally 8,9 and theoretically, 10–12 but most of 2 ACS Paragon Plus Environment

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the recent observations fail to detect molecular oxygen in the ISM. 13,14 By contrast, the hydroxyl radical OH is commonly observed in star-forming regions of the ISM. 15,16 The Si+OH reaction belongs to the class of reactions involving pairs of neutral radicals, which can generally proceed rapidly down to low temperatures due to the absence of potential energy barrier. 17,18 Kinetics experiments for this class of reactions are quite difficult to perform, and, to date, thermal rate constants could be measured in the temperature range relevant to the ISM only for the O+OH 19 and N+OH 20 reactions. Theoretical studies are thus highly needed to determine the kinetics data required for the modelling of the SiO chemistry in the ISM. The Si(3 P)+OH(X2 Π)→SiO(X1 Σ+ )+H(2 S) reaction is exoergic by 3.6 eV and can proceed adiabatically through the ground X 2 A′ electronic state. The first global potential energy surface (PES) for this state has been reported by some of us. 21 The PES is based on a large number of ab initio energies obtained from multireference configuration interaction calculations plus Davidson correction and basis sets of quadruple zeta quality. The PES incorporates an accurate description of long-range interactions in the reactant channel, determined from a multipolar expansion of the electrostatic interaction operator treated up to the second order of perturbation theory. 22 The analytical representation of the global PES was developed by means of the reproducing kernel Hilbert space method. The X 2 A′ PES was found barrierless, except for the approach of the reactants at nearly linear Si-HO geometries. The PES supports two deep energy wells along the reaction pathway, corresponding to the stable SiOH and HSiO isomers, with depths of 5.1 eV and 4.7 eV, respectively, relative to the Si+OH dissociation limit. The transition state energy barrier connecting the two isomers lies 3.5 eV below the reactants. Recently, we have reported the first theoretical study of the Si+OH reaction dynamics 23 based on the X 2 A′ PES. Total reaction probabilities, integral cross sections, state-selected and thermal rate constants were determined by means of the quasi-classical trajectory (QCT) method. 24 The reaction probabilities were found relatively low, due to significant backdissociation of the SiOH intermediate complex, and show a non-negligible dependence on both the collision energy and the initial rotational excitation of the OH molecule. When a thermal distribution over the reactants



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fine-structure levels is considered, the thermal rate constant reaches a maximum value of 4.3×10−10 cm3 .molecule−1 .s−1 at 20 K, followed by a rapid decrease with temperature. The aim of the present work was to perform a first quantum-mechanical (QM) study of dynamics of the Si+OH reaction, based on the same X 2 A′ PES. Owing to the large number of quantum states supported by the SiOH and HSiO wells, as well as the relatively large exoergicity of the system, an exact QM treatment of the reaction dynamics would be computationally very demanding. We thus chose to treat the reaction dynamics at a total angular momentum J = 0 by means of a time-dependent wave-packet (TDWP) approach. A flux analysis method has been employed to compute the total reaction probabilities for collision energies ranging from 10−6 to 1 eV, and for the first fifteen rotational states j = 0 − 14 of OH(v=0, j). The reaction probabilities for J > 0 were evaluated by a J-shifting approach. The latter quantities were also determined from QCT calculations performed at fixed values of the total and rotational angular momenta, J and j. The TDWP reaction probabilities were employed to compute integral cross sections, state-selected and thermal rate constants for the temperature range 10-500 K. The TDWP results for the cross sections and rate constants are compared with the QCT results reported in our previous work. 23

2 Methods and computational details 2.1 Wave packet calculations The quantum dynamics study of the Si+OH reaction is performed using the TDWP approach. The reactant Jacobi coordinate system (R,r, θ) is used, where R is the distance between the Si atom and the center-of-mass of the OH diatom, r is the internuclear distance of OH, and θ is the angle between the vectors R and r. The z-axis of the body-fixed (BF) frame is chosen to be parallel to R with the diatom in the xz plane. The TDWP calculations are carried out for a total angular momentum J = 0. The reaction probabilities for J > 0 are evaluated by a J-shifting approach 25,26


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briefly described in Sec. 2.2. The Hamiltonian operator for J = 0 writes as: ! 2 2 2 2 ~ ∂ ∂ ~ 1 1 ˆj2 + V(r, R, θ), Hˆ = − − + + 2µR ∂R2 2µr ∂r2 2µR R2 2µr r2

(1)

where µR = mS i (mO + mH )/(mS i + mO + mH ) and µr = mO mH /(mO + mH ) are the reduced masses for Si-OH and OH, respectively, ˆj is the rotational angular momentum operator of OH, and V(R, r, θ) is the PES for the ground X 2 A′ electronic state of the SiOH system. 21 The initial wave packet is built as the product of an incoming Gaussian wave packet (GWP) along R and a given rovibrational state of the reactant diatom:

ΨvJ=0 j (R, r, θ, t = 0) = G(R) ϕv j (r) P j (cos θ),

(2)

where ϕv j (r) is the eigenfunction for OH in its rovibrational (v, j) state, and P j (cos θ) is a normalized Legendre polynomial. The rovibrational wave function ϕv j (r) and its associated energy ǫv j are determined by the Fourier Grid Hamiltonian method. 27 The form used for G(R): 1 G (R) = 2πσ2

!1/4

# (R − R0 )2 − ik0 (R − R0 ) , exp − 4σ2 "

(3)

i h describes a GWP centered at R0 , with a width σ and a mean kinetic energy E0 = (~2 /2µR ) k02 + 1/4σ2 . The time propagation of the wave packet:

ΨvJ=0 j (R, r, θ, t

# ˆ −iH∆t + ∆t) = exp ΨvJ=0 j (R, r, θ, t) ~ "

(4)

is achieved using the split-operator method, 28,29 where the propagation time is discretized with uniform time steps ∆t. The action of the Hamiltonian onto the wave packet is computed using local representations of each operator, namely, a discrete variable representation 30–32 (DVR) for the potential energy operator, and a finite basis representation (FBR) for the radial and angular kinetic energy operators. The DVR is based on a direct product grid of NR × Nr × Nθ points,



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corresponding to equally spaced grid points for the radial coordinates R and r, and to GaussLegendre quadrature points for the angular coordinate θ. The basis set of the related FBR is defined as a direct product of NR × Nr plane waves and Nθ normalized Legendre polynomials. The kinetic energy terms are calculated in the FBR following DVR/FBR transformations of the wave packet, performed by two-dimensional fast Fourier transforms 33 (FFTs) for the radial grids, and an unitary matrix transformation for the angular grid. 30,34 A complex absorbing potential (CAP) is employed to avoid unphysical reflections of the wave packet at the edge of the grids in r and R. The CAP writes as −iWR (R)Wr (r), and is applied at each time step for R ≥ RCAP and r ≥ rCAP . We chose in this work the polynomial form proposed by Riss and Meyer, 35 W x (x) = η x (x − xCAP )3 , with x = R or r, where η x is a strength parameter. The final analysis of the wave packet is performed using a flux-based approach. 36–38 For each selected (v, j) state of OH, the total reaction probability PvJ=0 j (E), summed over all final states of the product SiO molecules, is calculated as the energy-dependent flux through a dividing surface placed in the product channel:

E D J=0 J=0 ˆ (E) (E) (E) , Φ F(r ) PvJ=0 = Φ d vj j vj

+# "* ~ ∂ J=0 J=0 = Im Φv j (E) δ(r − rd ) Φv j (E) , (5) µr ∂r

ˆ d ) is the quantum flux operator for the dividing surface defined by r = rd . The timewhere F(r J=0 independent (TI) scattering wave function ΦvJ=0 j (E) is obtained from the TD wave packet Ψv j (t)

by a Fourier transform:

ΦvJ=0 j

1 (R, r, θ, E) = av j (E)

Z

+∞

eiEt/~ ΨvJ=0 j (R, r, θ, t) dt.

(6)

−∞

The normalization coefficient av j (E) corresponds 37–39 to the energy amplitude contained in the


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initial wave packet for an incoming free wave φvJ=0 j (E) at energy E. Writing the free wave as:

φvJ=0 j (R, r, θ, E)

µR = ~kv j

!1/2

exp(−ikv j R) ϕv j (r) P j (cos θ),

(7)

D E J=0 where ~2 kv2 j /2µR = (E − ǫv j ) is the collision energy, the coefficient av j (E) = φvJ=0 (E) Ψ (0) is | j vj then given by:

µR av j (E) = ~kv j

!1/2 Z+∞ G(R) eikv j R dR,

(8)

−∞

where G(R) is the GWP defined by Eq. 3. Table 1: Parameters used in the TDWP calculations to get converged total reaction probabilities over the ranges of collision energy [10−6 − 10−3 ] eV (values in parenthesis) and [10−3 − 1] eV. Variable NR / Nr / Nθ Rmin / Rmax (a0 ) rmin / rmax (a0 ) σ (a0 ) E0 (eV) R0 (a0 ) T (ps) ∆t (fs) ηR / ηr (a.u.) RCAP / rCAP (a0 ) rd (a0 )

Value 1024 / 96 / 70 0.94 / 33.84 0.94 / 11.15 0.12 (0.17) 0.3 (0.1) 26.65 50 (100) 0.1 fs 0.003 / 0.003 24.46 / 7.37 7.37

Description Number of DVR grid points Minimal and maximal R values Minimal and maximal r values Width of the GWP Mean kinetic energy of the GWP Center of the GWP Total propagation time Propagation time step Strength parameters for the CAP Location of the CAP along R and r Location of the dividing surface

The values of the parameters used in the TDWP calculations are collected in Table 1. In order to get converged reaction probabilities over the whole collision energy range investigated here, 10−6 − 1 eV, it was necessary to perfom separate calculations for the low (10−6 − 10−3 eV) and relatively high (10−3 − 1 eV) energy domains. The same parameters were used for each of the selected rotational j state of OH(v = 0, j). Due to the presence of two deep energy wells associated with the stable SiOH and HSiO species, quite large propagation times have been required for both the low and ”high” energy ranges. With the chosen parameters, only about 0.5 − 0.8% of the wave packet is still trapped into the wells at the end of the propagation for all calculations. The main 7 ACS Paragon Plus Environment


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issue in TDWP calculations for the low energy domain relates to the ability of the CAP to absorb the wave packet at the edge of the grids, since the associated de Broglie wavelengths are large in this case. Several convergence tests were performed by increasing the length of the absorbing grids and by varying the strength parameters ηr and ηR of the CAPs in the r and R directions. Additional convergence tests concerned the parameters of the GWP (R0 and σ) and the location of the analysis line (rd ). In each case the reaction probabilities were found almost identical (to a very high degree of accuracy) to those obtained using the parameters given in Table 1. A final test was performed by varying the two energy domains, from [10−6 eV , 10−3 eV] to [10−5 eV , 10−2 eV] and from [10−3 eV , 1 eV] to [10−4 eV , 10−1 eV]. Such a shift implies a common energy domain between 10−4 eV and 10−2 eV. The probabilities obtained from the two separate calculations were found almost identical for this common energy domain. All these tests indicate that the TDWP calculations are well converged in the low energy domain. The TDWP code used in this work has been developed in the Laboratoire de Physique des Lasers, Atomes et Molecules (PhLAM), Université Lille 1, by G. Péoux and M. Monnerville. 34,40

2.2 Quantum cross sections and rate constants The integral reaction cross section (ICS) out of an initial rovibrational (v, j) state of the reactant diatom can be written as: 41–45 J

max X π 1 (2J + 1) 2 min(J, j) + 1 PvJ j (E), σv j (E) = 2 kv j (2 j + 1) J=0

(9)

where ~2 kv2 j /2µR = (E − ǫv j ), and PvJ j (E) is the total reaction probability associated with a given value of the total angular momentum J, summed over all final states of the products, and averaged over the helicity states Ω of the reactants (Ω being the projection of Jˆ and ˆj onto the BF z-axis). For the title reaction, we know from earlier QCT calculations 23 that the number of partial waves Jmax necessary to get converged cross sections can be quite large (up to 270 for the present energy range). An explicit calculation of the individual reaction probabilities PvJ j (E) for J > 0 would


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be computationally very demanding, either in a rigorous treatment (including all possible helicity states Ω and Coriolis couplings between adjacent components Ω and Ω±1), or within more approximated treatments, such as the centrifugal sudden approximation 41 (neglecting Coriolis couplings). Instead, we chose to get a first estimate of the reaction probabilities for J > 0 using the capture model proposed by Gray et al. 26 The model is closely related to the J-shifting approximation 46 in the sense that it assumes that the reaction probabilities are a function of the excess energy relative to a given barrier height depending on J. Using the results obtained for PvJ=0 j (E) to define such a function, the reaction probabilities for J > 0 are then evaluated as: J E − E PvJ j (E) = PvJ=0 j shift ,

(10)

J J where Eshift is the barrier height. For the specific case of barrierless reactions, Eshift is defined within

the capture model as the height of the centrifugal energy barrier of an effective one-dimensional potential given by:

VvJj (R) = hv j |V| v ji +

~2 J(J + 1), 2µR R2

(11)

where hv j |V| v ji is the X 2 A′ PES averaged over the initial rovibrational state of the reactants |v ji = E E ϕv j P j . The J-shifting approach has been previously applied to treat the J > 0 case for a series

of reactions between open-shell atoms (C, 47,48 N, 20 O, 49–51 S 52 ) and the OH radical, which are all barrierless reactions proceeding through deep potential energy wells. The ICS values obtained by the use of Eqs. 9 and 10 are employed to calculate the state-selected rate constants kv j (T ) corresponding to a Maxwell distribution of the reactants relative velocity at the temperature T : 8kB T kv j (T ) = πµR

! 21

×

Z

∞ 0

! ! Ec Ec Ec d , exp − σv j (Ec ) kB T kB T kB T

(12)

where Ec = (E − ǫv j ) is the collision energy, and kB is the Boltzmann constant. The state-selected



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rate constants so obtained are used to compute the thermal rate constants k(T ) corresponding to a Boltzmann distribution of the rovibrational states OH(v, j) at the temperature T : ∞ ∞ ǫv, j 1 XX k(T ) = (2 j + 1) exp − Qv j (T ) v=0 j=0 kB T

!

kv j (T ),

(13)

where Qv j (T ) is the partition function of the rovibrational states. The TDWP calculations of the reaction probabilities PvJ=0 j (E) for the first fifteen rotational states j = 0 − 14 of OH(v = 0, j) over the collision energy range 10−5 − 1 eV allowed us to derive converged values of the rate constants for the temperature range 10-500 K. The rate constants consistent with a thermal distribution of both the rovibrational states OH(v, j) and the fine-structure states Si(3 P J=0,1,2 ) and OH(2 ΠΩ=3/2,1/2 ) of the reactants are obtained multiplying k(T ) by an electronic partition function fe (T ). In that case, we neglect the changes induced by the fine-structure splittings on the long-range adiabatic PESs, 22 and assume that the reaction proceeds adiabatically on the X 2 A′ PES correlating with the lowest fine-structure level 53 Si(3 P0 )+OH(2 Π3/2 ). The temperature-dependent factor fe (T ) writes accordingly as:

g(2 A′ ) fe (T ) = 3 g( P0 ) + g(3 P1 ) exp (−∆E1/T ) + g(3 P2 ) exp (−∆E2/T ) ×

g(2 Π3/2 )

1 +

, (14)

g(2 Π1/2 ) exp (−∆EOH/T )

where g(2 A′ ) = 2 is the degeneracy of the X 2 A′ state due to spin multiplicity, g(3 P J ) = (2J + 1) is the degeneracy of each fine-structure level Si(3 P J ), and g(2 Π1/2 ) = g(2 Π3/2 ) = 2 is the degeneracy of the fine-structure levels OH(2 ΠΩ ). The fine-structure energy splittings were taken equal to their experimental values, 54,55 ∆E1 = 111 K, ∆E2 = 321 K, and ∆EOH = 201 K. The factor fe (T ) varies between 1 and 1/18 on increasing the temperature.


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2.3 QCT calculations The QCT results presented in this work for the state-selected integral reaction cross-sections (ICSs) and rate constants have been already reported in a previous publication 23 where all details of the calculations can be found. Briefly, batches of 105 trajectories were run for each collision energy Ec and rovibrational state OH(v = 0, j) following a standard Monte Carlo sampling 24 of the initial conditions, yielding ICS values σv j (Ec ) with statistical deviations of less than 1%. The calculations were repeated at selected energies in the collision energy range 10−3 − 1 eV for the first fifteen rotational states j = 0 − 14 of OH(v = 0, j). The energy dependence of each ICS σv j (Ec ) was fitted to an analytical form, which was then employed to compute the state-selected rate-constants kv j (T ) through Eq. 12. The QCT thermal rate constants k(T ) and fe (T )k(T ) correspond to Eqs. 13 and 14. For the present work, additional QCT calculations have been performed to determine the stateselected reaction probabilities PvJ j (Ec ) at fixed values of the total angular momentum J. In order to compare the TDWP and QCT results for any values of the angular momenta J and j, we employed the sampling scheme proposed by Aoiz et al. 56 to define the initial conditions of the trajectories. In that case, for fixed J and j values, the integer value of the orbital angular momentum l is sampled uniformly in the range |J − j| ≤ l ≤ J + j consistent with the vector addition J = j + l. The impact parameter is fixed by b = |l|/(2µR Ec )1/2 , where the quantization |l| = [l(l + 1)]1/2 ~ is applied, instead of being sampled (not uniformly) in the range 0 ≤ b ≤ bmax as in the standard approach (bmax being the maximum impact parameter beyond which no reaction occurs). The energy dependence of each reaction probability PvJ j (Ec ) was determined by running batches of 5 ×104 trajectories with the collision energy uniformly sampled in the range 10−3 − 1 eV. The initial atom-diatom separation was set to 40 a0 . The set of Hamilton equations is integrated using the step adaptive Runge-Kutta algorithm 57 with a relative precision of 10−8 for the distances and momenta, yielding to a conservation of the total energy and total angular momentum with an average error of 10−9 eV and 10−10 ~, respectively. The trajectories are stopped when the separation between the final fragments is greater than 20 a0 and their recoil energy varies by less than 10−7 eV between two consecutive time steps. The reaction probability PvJ j (Ec ) is obtained by the method of moment 11 ACS Paragon Plus Environment


1.0 J=0, OH(v=0,j=0)

0.8 Reaction probability

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QCT TDWP

0.6

0.4

0.2

0.0 0

0.2

0.4 0.6 Collision energy (eV)

0.8

1

Figure 1: Total reaction probability as a function of collision energy calculated by the TDWP (red line) and QCT (black line with error bars in grey) methods for the Si+OH(v=0, j=0) reaction at total angular momentum J = 0. expansion in Legendre polynomials. 24,56,58 The truncation of the polynomial expansion is chosen applying the Smirnov-Kolmogorov test. 57 For the cases considered in Sec. 3.1, the inclusion of 6-10 Legendre moments was sufficient to achieve significance levels higher than 95%.

3 Results and Discussions 3.1 Reaction probabilities The TDWP and QCT reaction probabilities for the Si+OH(v = 0, j = 0) → SiO+H reaction at total angular momentum J = 0 are shown in Fig. 1 as functions of the collision energy. The reaction is exothermic and the PES is barrierless in the reactant channel, except for a small barrier at linear Si-HO geometries. 21 In the absence of centrifugal energy barrier there is thus no energy threshold to the reaction. Regarding the energy dependence of the TDWP probability, we can distinguish three regions. In the low collision energy range Ec ≤ 0.45 eV (region I) the probability displays dense resonance structures. Some sharp peaks are relatively intense, in particular at the lowest energies, but the reaction probability is rather small on average (around 0.2). At intermediate


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energies 0.45 ≤ Ec ≤ 0.6 eV (region II), both the reaction probability and the amplitude of the resonance peaks gradually decrease on increasing energy. At higher energies Ec ≥ 0.6 eV (region III), the reaction probability becomes nearly constant (with an average value around 0.05) and the resonance structures are much broader. The sharp resonance peaks obssierved in the TDWP reaction probability are a signature of longlived quasibound states, supported by either one or the two deep wells of the PES corresponding to the bent SiOH and HSiO stable isomers. The depths of the SiOH and HSiO wells are, respectively, 5.1 eV and 4.7 eV relative to the Si+OH reactant channel, and the transition state energy barrier connecting them (referred as TS1 in ref. 21 ) lies 3.5 eV below the reactants. Only the SiOH well is accessible from the reactants, whereas both the SiOH and HSiO wells are connected to the products SiO+H through transition state energy barriers (referred as TS3 and TS6 for SiOH and HSiO, respectively) lying more than 3.2 eV below the reactants. A complete description of the stationary points of the PES can be found in Fig. 9 of ref. 21 The reaction can thus proceed solely via the SiOH well, or by an indirect pathway involving the two energy wells. Owing to the large exoergicity of the reaction, 3.6 eV, there is a huge number of product channels that are accessible, even at the lowest collision energies where all channels SiO(v′ ≤ 28, j′ )+H are open. The sharp resonances observed (region I) can thus not be explained in terms of a slow decay of the quasibound states due to the limited number of available product channels. 59 The resonance structures shown in Fig. 1 are similar to those obtained for the C+OH 60,61 (in its first and second excited states), N+OH, 62,63 O+OH 49,50 and S+OH 64 reactions, characterized by much smaller exoergicities. The quite low reaction probability indicates that the SiOH intermediate complex dissociates primarily back to the reactants, even though an excess energy of more than 3 eV is available to overcome the transition state energy barriers. Following the analysis proposed for the O+OH reaction, 65 this may be explained by an inefficient transfer of the collision energy to the O-H vibrational motion of the SiOH intermediate complex. For the present system, both the dissociation SiOH→TS3→SiO+H and isomerization SiOH→TS1→HSiO processes involve an increase of the O-H bond of SiOH, by at least 0.8 a0 according to the geometries of the two transition states. 21 The



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PES features hindering this energy transfer are the conservation of the O-H equilibrium distance along the minimum energy path leading to form SiOH (attack of Si to the O-side of OH), and the large frequency mismatch between the Si-O and O-H vibrational modes of SiOH, with ωSiO ≪ ωOH . In the particular case of the isomerization process, which involves a significant decrease of the ∠SiOH angle, a fast energy transfer to the SiOH bending mode is also required. At low collision energy, the small amount of energy channeled into SiOH internal modes can result in a loss of recoil energy sufficient to produce a long-time trapping of the system, with a large amplitude motion of Si relative to OH. This could explain the sharp resonance structures observed in the TDWP probability. Assuming that energy transfers to the internal modes do not improve with collision energy, no such trapping would occur at high collision energies, since it would be easier for the SiOH intermediate complex to dissociate back to the reactants. The absence of resonance structures as well as the lower reaction probability observed at high collision energies may be understood along these lines. As can be seen in Fig. 1, the QCT reaction probability at J = 0 coincides with the average value of the TDWP probability for Ec ≥ 0.3 eV, but it is found significantly larger than the TDWP result in the low collision energy range. Similar trends have been reported for the C+OH, 60,61 N+OH, 66 S+OH 64 and O+OH 67 reactions, although in that latter case the overestimate of the reaction probability by the QCT approach is much less pronounced. Leaving aside the resonance features, a possible source of discrepancies between the QCT and average TDWP results is the lack of zeropoint energy (ZPE) constraints in the QCT method. Indeed, in classical simulations the energy can flow between the different modes of motion without restriction on the ZPE that each mode must contain. Depending on the coupling terms, some modes may lose energy less than the ZPE, and the release of this amount of energy into other modes can affect the reaction dynamics. Several methods 68–75 have been proposed to enforce ZPE constraints in QCT calculations. We employed in this work the standard QCT procedure, where the ZPE of the reactant diatom OH(v = 0) (about 0.23 eV) is accounted for in the total energy, and all trajectories are included in the statistical analysis regardless of the ZPE requirement for the diatom in the final arrangement. In the case of Fig. 1,


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we found about 50% of the trajectories ending with diatoms having vibrational energy below the ZPE, independently of the collision energy. As expected from the large reaction exoergicity, most of them correspond to nonreactive events. Excluding from the analysis all trajectories that violate the ZPE constraint for the final diatom would result in a significant increase of the QCT reaction probability, and thus to larger discrepancies with the TDWP results. However, such a correction scheme (corresponding to the ZP3 procedure of Nyman and Davidsson 71 ) is known to introduce a bias in the Monte Carlo sampling, 65,71,76 and it ignores the ZPE constraints during the course of the reaction. The leak of ZPE from vibrational modes perpendicular to the reaction coordinate can allow the classical trajectories to follow reaction pathways that are inhibited for the quantum wavepacket. This may artificially enhance the probability for the SiOH complex to dissociate to the products, or isomerize to HSiO. However, the amount of ZPE here is small compared to the excess energy available to cross the energy barriers (more than 3 eV for TS3 and TS1). Even though the system may probe regions of the PES away from the minimum energy path, where the excess energy can be significantly smaller, further studies will be necessary to firmly attribute to ZPE effects the discrepancies between the TDWP and QCT results at low collision energy. Fig. 2 displays the TDWP and QCT reaction probabilities calculated at J = 0 as a function of collision energy for three selected excited rotational states ( j = 1,7,14) of OH(v = 0, j). Compared to the j = 0 case (Fig. 1), the resonance structures in the TDWP probability are still present and extend on a similar collision energy range, but the peaks become broader on increasing j values, possibly indicating a faster decay of the associated quasibound states. At low collision energies (region I), the average TDWP probability is almost independent of j up to j = 10 (not shown here), and, for j ≥ 10, an enhancement of the reaction probability by about a factor of two is observed. At high collision energies (region III), the TDWP probability increases continuously with j, up to a factor of six for j = 14 compared to j = 0. Regarding the QCT probability, the first salient feature is a significant decrease of the reaction probability at low energies going from j = 0 to j = 1. For j > 0 an overall good agreement is attained with the TDWP results over the whole energy range, apart from a slight underestimate of the TDWP probability for high j states.



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QCT TDWP

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Figure 2: Total reaction probabilities as a function of collision energy calculated by the TDWP (red line) and QCT (black line with error bars in grey) methods at total angular momentum J = 0 for three selected rotational states of OH(v=0, j). The main effect of the rotational excitation of OH(v = 0, j) is thus an enhancement of reactivity, from j ≥ 10 at low collision energies, and from j > 0 at high energies. Given the large amount of excess energy available to the reaction, the influence of rotational excitation on reactivity can 16 ACS Paragon Plus Environment

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be interpreted mainly in terms of orientational effects. At J = 0 the collision occurs in the plane defined by the free rotating molecule, with j = −l vectors being perpendicular to this plane. For j = 0 the collision follows an head-on approach of the fragments (zero impact parameter), whereas for j > 0 the approach is sideways, with impact parameters decreasing with collision energy as b = |l|/(2µR Ec )1/2 . Since the PES is barrierless for almost all angles of approach, the combined rotational and orbital motions lead the system to explore a wider range of geometries for j > 0 than at j = 0. If those geometries are favorable for energy transfers to the O-H vibrational mode, or the SiOH bending mode, an enhancement of reactivity is then expected. The decrease of the QCT probability going from j = 0 to j = 1 suggests that, at j = 0, a large part of reactive trajectories follows a pathway (inhibited for the quantum wavepacket) corresponding, at some point between the reactants and products, to a narrow range of angular geometries. With increasing rotation, or collision energy, it becomes increasingly difficult for the anisotropy of the PES to drive the fragments towards this particular orientation, giving rise to a decrease of reactivity. We compare in Fig. 3 the TDWP and QCT reaction probabilities calculated at fixed values of the total angular momentum J and for selected rotational states of OH(v = 0, j). The TDWP results for J > 0 were obtained using the J-shifting approach described in Sec. 2.2. Since the QCT approach restricts to classical dynamics, such a comparison can not serve as a rigorous test of the J-shifting approximation. However, it can serve to trace back the features observed for the integral cross sections and rate constants. As can be seen, the bump observed in the QCT probability at low energy for j = 0 and J = 0 (Fig. 1) is still present for J > 0. This provides another clue for the contribution of a particular reaction pathway at j = 0, disrupted only by the rotational motion of OH. We also observe that the QCT probabilities calculated at large J values are smaller than at J = 0, the effect being more pronounced at high energy and large j values. This may be understood by the increasing difficulty for the interaction potential to capture an incoming Si atom associated with large orbital motion (since |J − j| ≤ l ≤ J + j), and the dependence on j of the capture process can be related to the anisotropic character of the potential in the reactant channel.



1.0 J=50, OH(v=0,j=0)


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QCT TDWP

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Figure 3: Total reaction probabilities as a function of collision energy calculated by the TDWP (red line) and QCT (black line with error bars in grey) methods for selected rotational states of OH(v=0, j) and fixed values of the total angular momentum J. The TDWP results are obtained within the J-shifting approximation.


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3.2 Integral cross sections and rate constants 250 QCT TDWP

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Figure 4: Integral cross sections as a function of collision energy calculated by the TDWP (red line) and QCT (black line) methods for the Si+OH reaction for three selected rotational states of OH(v=0, j). The TDWP results for the integral cross sections (ICSs) and rate constants were obtained as 19 ACS Paragon Plus Environment


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described in Sec. 2.2 from the quantum reaction probabilities previously discussed in Sec. 3.1. The QCT results presented below are reproduced from a previous work 23 on the title reaction. The TDWP and QCT ICSs are displayed in Fig. 4 as a function of collision energy for three selected rotational states ( j = 0,7,14) of OH(v=0, j). The ICSs are monotonically decreasing functions of collision energy, what is the expected feature for barrierless reactions. At low collision energy, the TDWP cross sections exhibit structures which are due to the sharp resonance peaks observed in the quantum reaction probabilities (Figs. 1-3). Since the resonance peaks become slightly broader with increasing OH rotation, the structures in the TDWP ICSs gradually disappear for high j states. For collision energies below 0.2 eV, the QCT ICSs are larger than the TDWP ones for all rotational states OH(v=0, j) with j ≤ 7. The greatest discrepancies are observed for the j = 0 case, where they can rise up to a factor of two at about 0.1 eV. This is a consequence of the large differences observed between the TDWP and QCT reaction probabilities for OH(v=0, j=0) in the low energy range. Since this unwanted feature tends to disappear with the initial OH rotational motion, the discrepancies between the TDWP and QCT ICSs significantly reduce for the excited rotational states OH(v=0, j). Above 0.2 eV, the TDWP ICSs are larger than the QCT ones for all j values, as expected from the features observed for the corresponding reaction probabilities. Regarding the dependence of the ICSs on the initial rotational state OH(v=0, j), both the TDWP and QCT approaches predict similar trends, with an increase of the ICS values with increasing j from j ≥ 10 at low energy, and from j > 0 in the high energy range. Note that deviations from these general trends are observed for some particular j states and collision energy values. We report in Fig. 5 the TDWP and QCT state-selected rate constants kv j (T ) as a function of temperature for the same three rotational states OH(v=0, j) considered in Fig. 4. We recall that, for the temperature range 10-500 K, the ICSs for collision energies below 0.2 eV provide the main contribution to the rate constant values. Accordingly, the QCT rate constants are larger than the TDWP ones for all OH(v=0, j) states with j ≤ 7, and for the specific case j = 14. Large discrepancies are found for OH(v=0, j=0), up to a factor of two at 500 K, but they are significantly smaller for the excited rotational states. Both approaches give similar temperature dependences of


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Figure 5: State-selected rate constants as a function of temperature calculated by the TDWP (red line) and QCT (black line) methods for the Si+OH reaction for three selected rotational states of OH(v=0, j). the rate constants, with a rapid increase with T below ∼50 K, followed by a slow decrease with T for j < 10, or a slow increase for j ≥ 10. The general trend with j of the rate constant values



corresponds to an increase with increasing j from j ≥ 10, whatever the temperature range. A rapid decrease with j for j ≤ 3 (not shown here) is also obtained below 100 K from both approaches. 5 10-10 k(T) QCT k(T) TDWP

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Figure 6: Thermal rate constants as a function of temperature calculated by the TDWP (red line) and QCT (black line) methods for the Si+OH→SiO+H reaction, including (solid line) or not including (dashed line) the electronic partition function fe (T ) of Eq. 14 corresponding to a thermal distribution of the fine-structure levels of the reactants. In Fig. 6 are displayed the TDWP and QCT thermal rate constants k(T ), obtained from the state-selected rate constants kv j (T ) computed for the first fifteen rotational states of OH(v=0, j). Both thermal rate constants show a rapid increase with T up to a maximum value, at 15 K and 25 K for the TDWP and QCT results, respectively, followed by a slow decrease with increasing T . The rate constant values at the associated maxima, 3.7 × 10−10 cm3 .molecule−1 .s−1 and 4.4 × 10−10 cm3 .molecule−1 .s−1 , differ by about 20%. The Boltzmann distribution over the energy levels OH(v=0, j) shows that the states with j ≤ 7 provide the main contribution to the rate constant values up to 500 K, the rotational level j = 3 being the most populated at 500 K. Accordingly, the QCT results are larger than the TDWP ones over the whole temperature range, with discrepancies varying between 15% at 10 K to 50% at 500 K, and a maximal deviation of 70% at 200 K. The negative temperature dependence observed for k(T ) from both approaches is due to the increasing contribution of low-lying rotational states ( j ≤ 3) associated with decreasing kv j (T ) values. We also report in Fig. 6 the values obtained for the rate constants fe (T )k(T ), corresponding to a thermal distribution of populations over both the rovibrational levels OH(v, j) and the fine22 ACS Paragon Plus Environment

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structure levels Si(3 P J ) and OH(2 ΠΩ ) of the reactants (see Sec. 2.2). Owing to the relatively large values of the fine-structure splittings, the results are quite similar to k(T ) in the low temperature range. In particular, the location of the two maxima, and their associated rate constant values, are almost unchanged. The main effect of the electronic factor fe (T ) is a sharp decrease of the rate constant values on increasing T beyond ∼20 K, as a result of the increasing population in the nonreactive excited fine-structure levels. Consequently, the rate constants values fe (T )k(T ) vary by more than one order of magnitude between the location of the maxima (15 K and 20 K for the TDWP and QCT results, respectively) and the high temperature range. This is quite different from the temperature independent value of 10−10 cm3 s−1 reported in astrochemical databases 77–79 for the rate constant of the Si+OH reaction. In order to provide a thermal rate constant suitable for use in chemical models, we chose to fit the TDWP values of fe (T )k(T ) to the standard Arrhenius-Kooij functional form: kstandard (T ) = α

T β γ exp − . 300 T

(15)

Two distinct set of parameters were found necessary to properly reproduce the TDWP results over the whole temperature range. Using least-square procedures, we found α = 3.04 × 10−11 cm3 .molecule−1 .s−1 , β = −1.34 and γ = 21.98 K for T ≤ 300 K, and α = 3.15 × 10−11 cm3 s−1 , β = −0.59 and γ = 0 K for T > 300 K. The maximal deviations between the analytical form and the TDWP values are 5% and 2% for the low and high temperature ranges, respectively. The parameters corresponding to the QCT results are available in Ref. 23 Future works using astro-chemical models that include the Si+OH reaction must be performed in order to determine the impact of the updated rate constant values on the prediction of astrophysical quantities. Several studies, based on the Monte Carlo method, 80 have been devoted to determine the sensitivity of the predicted molecular abundances to the uncertainties of the reaction rate constants included in chemical models. The improved accuracy of the rate constant values serve to better constrain such models. 81 In the absence of experimental data, and compared to the constant value of 10−10 cm3 .molecule−1 .s−1 currently employed in chemical models, the rate constant values reported in this work are thought to be the most appropriate for the modelling of the 23 ACS Paragon Plus Environment


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SiO chemistry in many astrophysical environments.

4 Conclusions We have reported a first quantum study of the dynamics of the Si(3 P)+OH(X2 Π) → SiO(X1 Σ+ )+H(2 S) reaction, based on TDWP calculations and a PES associated with the ground electronic state X 2 A′ of the system. 21 Total reaction probabilities have been computed at total angular momentum J = 0 for the first fifteen rotational states of OH(v=0, j) over the collision energy range 10−6 -1 eV. At low collision energies (below 0.5 eV), the TDWP probabilities show dense resonance structures associated with long-lived quasi-bound states supported by the SiOH and HSiO deep wells of the PES. The average values of the TDWP probabilities are generally small (below 0.5). This has been attributed to backdissociation of the SiOH intermediate complex, due to inefficient energy transfers to the internal modes of SiOH that hinder the SiOH→SiO+H and SiOH→HSiO→SiO+H reaction pathways. The rotational excitation of OH(v=0, j) leads to an enhancement of reactivity, from j ≥ 10 at low collision energies, and from j > 0 at high energies. The QCT probabilities computed at J = 0 globally agree with the TDWP results. A notable exception is the OH(v=0, j=0) case, for which the QCT results overestimate significantly the reaction probability at low energies (around 0.1 eV). A possible explanation is a leak of ZPE during the course of classical trajectories, what can allow a reaction pathway that is inhibited for the quantum wavepacket. Below 0.2 eV, the TDWP integral cross sections are smaller than the QCT ones for all rotational states OH(v=0, j) with j ≤ 7. As a result, the TDWP state-selected and thermal rate constants are smaller than the QCT results for the whole temperature range 10-500 K. When a thermal distribution of populations over the reactants fine-structure levels is considered, the TDWP thermal rate constant reaches a maximum value of 3.6 × 10−10 cm3 .molecule−1 .s−1 at 15 K, followed by a rapid fall with temperature by one order of magnitude up to ∼200 K. This result differs significantly from the temperature-independent value currently reported in astrochemical databases. We proposed here a new analytical representation of the thermal rate constant, based on the standard Arrhenius-Kooij


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formula, well-suited to the modelling of the SiO chemistry in astrophysical environments. A detailed study of the state-to-state reaction dynamics will be presented in a forthcoming publication.

5 Acknowledgments The Laboratoire de Physique des Lasers, Atomes et Molécules is unité associée au CNRS, UMR ´ 8523. The ”Centre d’Etude et de Recherche Lasers et Applications” (CERLA) is supported by the French Ministry of Higher Education and Research, the European Regional Development Fund (ERDF) and the Region ”Les Hauts de France”. This work was also supported by the Embassy of France in Cuba and the CNRS national programme ”Physique et Chimie du Milieu Interstellaire”.

References (1) Mccarthy, M.; Gottlieb, C.; Thaddeus, P. Silicon molecules in space and in the laboratory. Mol. Phys. 2003, 101, 697–704. (2) Martin-Pintado, J.; Bachiller, R.; Fuente, A. SiO emission as a tracer of shocked gas in molecular outflows. Astron. Astrophys. 1992, 254, 315. (3) Codella, C.; Bachiller, R.; Reipurth, B. Low and high velocity SiO emission around young stellar objects. Astron. Astrophys. 1999, 343, 585. (4) Gusdorf, A.; Cabrit, S.; Flower, D. R.; des Forêts, G. P. SiO line emission from C-type shock waves: interstellar jets and outflows. Astron. Astrophys. 2008, 482, 809–829. (5) López-Sepulcre, A.; Watanabe, Y.; Sakai, N.; Furuya, R.; Saruwatari, O.; Yamamoto, S. The role of SiO as a tracer of past star-formation events: The case of the high-mass protocluster NGC 2264-C. Astrophys. J. 2016, 822, 85. (6) Langer, W.; Glassgold, A. Silicon chemistry in interstellar clouds. Astrophys. J. 1990, 352, 123–131. 25 ACS Paragon Plus Environment


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(7) Schilke, P.; Walmsley, C.; Pineau des Forets, G.; Flower, D. SiO production in interstellar shocks. Astron. Astrophys. 1997, 321, 293–304. (8) Le Picard, S.; Canosa, A.; des Forêts, G.; Rebrion-Rowe, C.; Rowe, B. The Si (P)+ O2 reaction: A fast source of SiO at very low temperature; CRESU measurements and interstellar consequences. Astron. Astrophys. 2001, 372, 1064–1070. (9) Gomez Martin, J. C.; Blitz, M. A.; Plane, J. M. Kinetic studies of atmospherically relevant silicon chemistry Part I: Silicon atom reactions. Phys. Chem. Chem. Phys. 2009, 11, 671–678. (10) Le Picard, S. D.; Canosa, A.; Reignier, D.; Stoecklin, T. A comparative study of the reactivity of the silicon atom Si(3 P J ) towards O2 and NO molecules at very low temperature. Phys. Chem. Chem. Phys. 2002, 4, 3659–3664. (11) Dayou, F.; Spielfiedel, A. Ab initio calculation of the ground (1 A′ ) potential energy surface and theoretical rate constant for the Si+ O → SiO+ O reaction. J. Chem. Phys. 2003, 119, 4237. (12) Dayou, F.; Larrégaray, P.; Bonnet, L.; Rayez, J.; Arenas, P.; González-Lezana, T. A comparative study of the Si+ O → SiO+ O reaction dynamics from quasiclassical trajectory and statistical based methods. J. Chem. Phys. 2008, 128, 174307. (13) Goldsmith, P. F.; Liseau, R.; Bell, T. A.; Black, J. H.; Chen, J.-H.; Hollenbach, D.; Kaufman, M. J.; Li, D.; Lis, D. C.; Melnick, G. et al. Herschel measurements of molecular oxygen in Orion. Astrophys. J. 2011, 737, 96. (14) Wirström, E. S.; Charnley, S. B.; Cordiner, M. A.; Ceccarelli, C. A search for O2 in COdepleted molecular cloud cores with Herschel. Astrophys. J. 2016, 830, 102. (15) Goicoechea, J. R.; Cernicharo, J.; Lerate, M. R.; Daniel, F.; Barlow, M. J.; Swinyard, B. M.; Lim, T. L.; Viti, S.; Yates, J. Far-Infrared excited hydroxyl lines from Orion KL outflows. Astrophys. J., Lett. 2006, 641, L49. 26 ACS Paragon Plus Environment

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(16) Wampfler, S. F.; Bruderer, S.; Karska, A.; Herczeg, G. J.; van Dishoeck, E. F.; Kristensen, L. E.; Goicoechea, J. R.; Benz, A. O.; Doty, S. D.; McCoey, C. et al. Herschel observations of the hydroxyl radical (OH) in young stellar objects. Astron. Astrophys. 2013, 552, A56. (17) Smith, I. W. M. Laboratory studies of atmospheric reactions at low temperatures. Chem. Rev. 2003, 103, 4549–4564. (18) Smith, I. W. Laboratory astrochemistry: Gas-phase processes. Annu. Rev. Astron. Astrophys. 2011, 49, 29–66. (19) Carty, D.; Goddard, A.; Khler, S. P. K.; Sims, I. R.; Smith, I. W. M. Kinetics of the radicalradical reaction, O(3 P J ) + OH(X2 ΠΩ ) → O2 + H, at temperatures down to 39 K. J. Phys. Chem. A 2006, 110, 3101–3109. (20) Daranlot, J.; Jorfi, M.; Xie, C.; Bergeat, A.; Costes, M.; Caubet, P.; Xie, D.; Guo, H.; Honvault, P.; Hickson, K. Revealing atom-radical reactivity at low temperature through the N + OH reaction. Science 2011, 334, 1538. (21) Dayou, F.; Duflot, D.; Rivero-Santamar´ıa, A.; Monnerville, M. A global ab initio potential energy surface for the X2 A′ ground state of the Si + OH → SiO + H reaction. J. Chem. Phys. 2013, 139, 204305. (22) Bussery-Honvault, B.; Dayou, F. Si + OH interaction: long-range multipolar potentials of the eighteen spin- orbit states. J. Phys. Chem. A 2009, 113, 14961–14968. (23) Rivero-Santamar´ıa, A.; Dayou, F.; Rubayo-Soneira, J.; Monnerville, M. Quasi-classical trajectory calculations of cross sections and rate constants for the Si + OH → SiO + H reaction. Chem. Phys. Lett. 2014, 610–611, 335–340. (24) D. G. Truhlar and J. T. Muckerman, In Atom-molecule collision theory; Bernstein, R. B., Ed.; Plenum, New York, Chapter 16, 1979. 27 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(25) Bowman, J. M. Reduced dimensionality theory of quantum reactive scattering. J. Phys. Chem. 1991, 95, 4960–4968. (26) K. Gray, S.; M. Goldfield, E.; C. Schatz, G.; G. Balint-Kurti, G. Helicity decoupled quantum dynamics and capture model cross sections and rate constants for O (1D) + H2 → OH + H. Phys. Chem. Chem. Phys. 1999, 1, 1141–1148. (27) Marston, C. C.; Balint-Kurti, G. G. The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions. J. Chem. Phys. 1989, 91, 3571–3576. (28) Feit, M.; Fleck, J.; Steiger, A. Solution of the Schrödinger equation by a spectral method. J. Comput. Phys. 1982, 47, 412–433. (29) Feit, M. D.; Fleck, J. A. Solution of the Schrdinger equation by a spectral method II: Vibrational energy levels of triatomic molecules. J. Chem. Phys. 1983, 78, 301–308. (30) Light, J. C.; Hamilton, I. P.; Lill, J. V. Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys. 1985, 82, 1400–1409. (31) Lill, J. V.; Parker, G. A.; Light, J. C. The discrete variable finite basis approach to quantum scattering. J. Chem. Phys. 1986, 85, 900–910. (32) Kosloff, R. Time-Dependent quantum-mechanical methods for molecular dynamics. J. Phys. Chem. 1988, 92, 2087–2100. (33) Kosloff, D.; Kosloff, R. A fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics. J. Comput. Phys. 1983, 52, 35 – 53. (34) Monnerville, M.; Péoux, G.; Briquez, S.; Halvick, P. Three-dimensional time-dependent study of a reaction involving three different heavy atoms and a very deep well: application to the C+NO → CN+O exchange reaction. Chem. Phys. Lett. 2000, 322, 157 – 165.


Page 28 of 36

Page 29 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(35) Riss, U. V.; Meyer, H.-D. The transformative complex absorbing potential method: a bridge between complex absorbing potentials and smooth exterior scaling. J. Phys. B: At., Mol. Opt. Phys. 1998, 31, 2279. (36) Miller, W. H. Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants. J. Chem. Phys. 1974, 61, 1823–1834. (37) Zhang, D. H.; Zhang, J. Z. H. Quantum reactive scattering with a deep well: Time-dependent calculation for H+O2 reaction and bound state characterization for HO2 . J. Chem. Phys. 1994, 101, 3671–3678. (38) Zhang, J. Z. H. Theory and application of quantum molecular dynamics, 1st ed.; World Scientific, Singapore, 1999. (39) Tannor, D. J.; Weeks, D. E. Wave packet correlation function formulation of scattering theory: The quantum analog of classical S matrix theory. J. Chem. Phys. 1993, 98, 3884–3893. (40) Péoux, G. Thése de l’Université de Lille I. Ph.D. thesis, 1997. (41) Pack, R. T. Space-fixed vs body-fixed axes in atom-diatomic molecule scattering. Sudden approximations. J. Chem. Phys. 1974, 60, 633–639. (42) Schatz, G. C.; Kuppermann, A. Quantum mechanical reactive scattering for three dimensional atom plus diatom systems. I. Theory. J. Chem. Phys. 1976, 65, 4642. (43) Zhang, J. Z. H.; Miller, W. H. Quantum reactive scattering via the S matrix version of the Kohn variational principle: Differential and integral cross sections for D+H2 → HD+H. J. Chem. Phys. 1989, 91, 1528. (44) Alagia, M.; Balucani, N.; Cartechini, L.; Casavecchia, P.; Volpi, G. G.; Javier Aoiz, F.; Banares, L.; Allison, T. C.; Mielke, S. L.; Truhlar, D. G. Dynamics of the Cl+H2 /D2 reaction: A comparison of crossed molecular beam experiments with quasiclassical trajectory and quantum mechanical calculations. Phys. Chem. Chem. Phys. 2000, 2, 599–612. 29 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(45) Aslan, E.; Bulut, N.; Castillo, J. F.; Baares, L.; Roncero, O.; Aoiz, F. J. Accurate TimeDependent wave packet study of the Li + H2 + reaction and Its isotopic variants. J. Phys. Chem. A 2012, 116, 132–138. (46) Sun, Q.; Bowman, J. M.; Schatz, G. C.; Sharp, J. R.; Connor, J. N. L. Reduced-dimensionality quantum calculations of the thermal rate coefficient for the Cl + HCl → ClH + Cl reaction: Comparison with centrifugal-sudden distorted wave, coupled channel hyperspherical, and experimental results. J. Chem. Phys. 1990, 92, 1677–1686. (47) Bulut, N.; Zanchet, A.; Honvault, P.; Bussery-Honvault, B.; Bañares, L. Time-dependent wave packet and quasiclassical trajectory study of the C + OH → CO + H reaction at the state-to-state level. J. Chem. Phys. 2009, 130, 194303. (48) Jorfi, M.; Honvault, P. Quantum dynamics at the state-to-state level of the C + OH reaction on the first excited potential energy surface. J. Phys. Chem. A 2010, 114, 4742–4747. (49) Xu, C.; Xie, D.; Honvault, P.; Lin, S. Y.; Guo, H. Rate constant for O+OH reaction on an improved ab initio potential energy surface and implications for the interstellar oxygen problem. J. Chem. Phys. 2007, 127, 024304. (50) Quéméner, G.; Balakrishnan, N.; Kendrick, B. K. Quantum dynamics of the O+OH → H+O2 reaction at low temperatures. J. Chem. Phys. 2008, 129, 0224309. (51) Lique, F.; Jorfi, M.; Honvault, P.; Halvick, P.; Lin, S.; Guo, H.; Xie, D.; Dagdigian, P.; Kłos, J.; Alexander, M. O + OH → O2 + H: A key reaction for interstellar chemistry. New theoretical results and comparison with experiment. J. Chem. Phys. 2009, 131, 221104. (52) Jorfi, M.; Honvault, P. Quantum dynamics of the S+OH → SO+H reaction. J. Chem. Phys. 2010, 133, 144315. (53) Graff, M. M.; Wagner, A. F. Theoretical studies of fine-structure effects and long-range forces: Potential-energy surfaces and reactivity of O + OH. J. Chem. Phys. 1990, 92, 2423. 30 ACS Paragon Plus Environment

Page 30 of 36

Page 31 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(54) National Institute of Standards and Technology, http://www.nist.gov/ (55) Chase, M. W.; Curnutt, J. L.; Downey, J. R.; McDonald, R. A.; Syverud, A. N.; Valenzuela, E. A. JANAF thermochemical tables. J. Phys. Chem. Ref. Data 1982, 11, 695. (56) Aoiz, F. J.; Sáez-Rábanos, V.; Mart´ınez-Haya, B.; González-Lezana, T. Quasiclassical determination of reaction probabilities as a function of the total angular momentum. J. Chem. Phys. 2005, 123, 094101. (57) Press, W. H.; Teukolsky, S. A.; Vetterling, V. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press, Cambridge, 1996. (58) Aoiz, F. J.; Herrero, V. J.; Rábanos, V. S. Quasiclassical state to state reaction cross sections for D+H2 (v=0, j=0) → HD(v′ , j′ )+H. Formation and characteristics of short-lived collision complexes. J. Chem. Phys. 1992, 97, 7423–7436. (59) Rackham, E. J.; Gonzalez-Lezana, T.; Manolopoulos, D. E. A rigorous test of the statistical model for atom diatom insertion reactions. J. Chem. Phys. 2003, 119, 12895–12907. (60) Zanchet, A.; Gonzalez-Lezana, T.; Roncero, O.; Jorfi, M.; Honvault, P.; Hankel, M. An accurate study of the dynamics of the C+OH reaction on the second excited 14 A” potential energy surface. J. Chem. Phys. 2012, 136, 164309. (61) Rao, T. R.; Goswami, S.; Mahapatra, S.; Bussery-Honvault, B.; Honvault, P. Time-dependent quantum wave packet dynamics of the C + OH reaction on the excited electronic state. J. Chem. Phys. 2013, 138, 094318. (62) Bulut, N.; Roncero, O.; Jorfi, M.; Honvault, P. Accurate time dependent wave packet calculations for the N + OH reaction. J. Chem. Phys. 2011, 135, 104307. (63) Jorfi, M.; Honvault, P. State-to-State quantum dynamics of the N + OH → NO + H reaction. J. Phys. Chem. A 2009, 113, 2316–2322.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(64) Goswami, S.; Rao, T. R.; Mahapatra, S.; Bussery-Honvault, B.; Honvault, P. Time-dependent quantum wave packet dynamics of the S + OH reaction on its electronic ground state. J. Phys. Chem. A 2014, 118, 5915–5926. (65) Jorfi, M.; Honvault, P.; no, P. B.; González-Lezana, T.; Larrégaray, P.; Bonnet, L.; Halvick, P. On the statistical behavior of the O + OH → H + O reaction: A comparison between quasiclassical trajectory, quantum scattering, and statistical calculations. J. Chem. Phys. 2009, 130, 184301. (66) Jorfi, M.; Honvault, P.; Halvick, P. Quasi-classical determination of integral cross-sections and rate constants for the N + OH → NO + H reaction. Chem. Phys. Lett. 2009, 471, 65 – 70. (67) Jorfi, M.; Honvault, P.; Halvick, P.; Lin, S.; Guo, H. Quasiclassical trajectory scattering calculations for the OH + O → H+ O2 reaction: Cross sections and rate constants. Chem. Phys. Lett. 2008, 462, 53–57. (68) Schatz, G. C. The origin of cross section thresholds in H+H2 : Why quantum dynamics appears to be more vibrationally adiabatic than classical dynamics. J. Chem. Phys. 1983, 79, 5386–5391. (69) Bowman, J. M.; Gazdy, B.; Sun, Q. A method to constrain vibrational energy in quasiclassical trajectory calculations. J. Chem. Phys. 1989, 91, 2859–2862. (70) Miller, W. H.; Hase, W. L.; Darling, C. L. A simple model for correcting the zero point energy problem in classical trajectory simulations of polyatomic molecules. J. Chem. Phys. 1989, 91, 2863–2868. (71) Nyman, G.; Davidsson, J. A low-energy quasiclassical trajectory study of O (3 P)+ OH (2Π) → O2 (3 Σg )+ H (2 S). II. Rate constants and recrossing, zero-point energy effects. J. Chem. Phys. 1990, 92, 2415–2422.


Page 32 of 36

Page 33 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(72) Varandas, A. J. C. Excitation function for H+O2 reaction: A study of zero point energy effects and rotational distributions in trajectory calculations. J. Chem. Phys. 1993, 99, 1076–1085. (73) Varandas, A. J. C.; Marques, J. M. C. Method for quasiclassical trajectory calculations on potential energy surfaces defined from gradients and Hessians, and model to constrain the energy in vibrational modes. J. Chem. Phys. 1994, 100, 1908–1920. (74) Bonnet, L.; Rayez, J.-C. Gaussian weighting in the quasiclassical trajectory method. Chem. Phys. Lett. 2004, 397, 106 – 109. (75) Paul, A. K.; Hase, W. L. Zero-Point energy constraint for unimolecular dissociation reactions. Giving trajectories multiple chances to dissociate correctly. J. Phys. Chem. A 2016, 120, 372– 378. (76) Varandas, A. J. C.; Marques, J. M. C. A detailed state to state low energy dynamics study of the reaction O(3 P) + OH(2Π) → O2 (X˜ 3 Σ−g ) + H(2S) using a quasiclassical trajectory internal energy quantum mechanical threshold method. J. Chem. Phys. 1992, 97, 4050–4065. (77) Smith, I. W. M.; Herbst, E.; Chang, Q. Rapid neutral-neutral reactions at low temperatures: a new network and first results for TMC-1. Mon. Not. R. Astron. Soc. 2004, 350, 323–330. (78) Woodall, J.; Agundez, M.; Markwick-Kemper, A. J.; Millar, T. J. The UMIST database for astrochemistry 2006. Astron. Astrophys. 2007, 466, 1197–1204. (79) Wakelam, V.; Herbst, E.; Loison, J.-C.; Smith, I. W. M.; Chandrasekaran, V.; Pavone, B.; Adams, N. G.; Bacchus-Montabonel, M.-C.; Bergeat, A.; Broff, K. et al. A kinetic database for astrochemistry (KIDA). Astrophys. J., Suppl. Ser. 2012, 199, 21. (80) Wakelam, V.; Loison, J.-C.; Herbst, E.; Talbi, D.; Quan, D.; Caralp, F. A sensitivity study of the neutral-neutral reactions C + C3 and C + C5 in cold dense interstellar clouds. Astron. Astrophys. 2009, 495, 513–521.



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(81) Wakelam, V.; Smith, I. W. M.; Herbst, E.; Troe, J.; Geppert, W.; Linnartz, H.; berg, K.; Roueff, E.; Agúndez, M.; Pernot, P. et al. Reaction networks for interstellar chemical modelling: Improvements and challenges. Space Sci. Rev. 2010, 156, 13–72.


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