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A quantum-classical study of the OH + H 2 reactive and

inelastic collisions

Carles Martí Aliod, Leonardo Pacifici, Antonio Lagana, Cecilia Coletti

To cite this version:

Carles Martí Aliod, Leonardo Pacifici, Antonio Lagana, Cecilia Coletti. A quantum-classical study of the OH + H 2 reactive and inelastic collisions. Chemical Physics Letters, Elsevier, 2017, 674, pp.103 - 108. �10.1016/j.cplett.2017.02.040�. �hal-01862548�

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A quantum-classical study of the OH + H

2

reactive and

inelastic collisions

Carles Mart´ı*, Leonardo Pacifici, Antonio Lagan`a

Dipartimento di Chimica, Biologia e Biotecnologie, Universit`a di Perugia, via Elce di Sotto, 06123 Perugia, Italy

Cecilia Coletti

Dipartimento di Farmacia, Universit`a G. d’Annunzio Chieti-Pescara, via dei Vestini, 66100 Chieti, Italy

Abstract

We carried out a study of OH + H2 scattering using a quantum-classical

method, treating quantally vibrations and classically both translations and rotations. The good agreement between the state specific quantum-classical reactive probabilities and the corresponding quantum ones prompted the ex-tension of the study to state to state probabilities for non reactive vibrational energy exchange. The study showed that H2 reactive dynamics depends on

the vibrational excitation, while the non reactive one is mainly vibrationally adiabatic. On the contrary, OH reactive dynamics is not affected by its vibra-tional excitation, whereas the non reactive one might produce some pumping up to higher vibrational states.

Keywords: Quantum-Classical methods, reactive probabilities, state to state non reactive probabilities, vibrational

adiabaticity

Corresponding author.

E-mail adress: carles.marti2@gmail.com (Mart´ı, C.)

This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

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1. Introduction The OH (2Π) + H

2 (1Σ+g) → H2O (X1A1) + H(2S) reaction together

with its isotopic variants is an important case study of four atom elementary process because of the role played in combustion and atmospheric chemistry [1, 2]. This reaction represents, as well, one of the simplest and prototypical

5

cases of four body processes, due to the fact that three out of the four in-tervening atoms are hydrogens with the consequent simplicity of electronic structure calculation and fitting to a reliable Potential Energy Surface (PES). Moreover, the lightness of H exalts quantum effects and makes the compari-son of experimental work (based on flash photolysis [3, 4] and crossed beams

10

[5, 6]) with theoretical studies [7, 8, 9] particularly interesting.

As a matter of fact, for this system a certain number of ab initio PES’s have been produced over the years [10, 11, 12, 13]. Full quantum techniques have also been applied to dynamical studies of this process both for J = 0 and for J > 0 [7, 13, 14, 15] and for its isotopic variants [16, 17]. Full

15

quantum calculations at J > 0 demand an increasingly heavier amount of both compute time and memory resources as the total angular momentum J increases, making them very costly. A way of carrying out a full dimensional study without sacrificing the possible quantum features of vibrational energy exchanges and maintaining, at the same time, reasonable time and resources

20

expense is to adopt a mixed Quantum-Classical (QC) technique[8, 18, 19]. In the QC method the quantum and classical treatments deal separately with different degrees of freedom whose coupling is ensured via an effective Hamiltonian formulated in terms of the expectation value of the semiclassical Hamiltonian, computed using the quantal wavefunction.

25

As discussed in this paper the recent publication of an ab initio potential energy surface of the title system by Chen et al.[13] and of the related full quantum estimates of state specific probabilities has allowed us both to test the accuracy of the QC treatment[20] and to extend the study to the state to state exchange of vibrational energy. Accordingly, the paper is organized as

30

follows: in Section 2 the adopted QC method is described; in Section 3 details on the performed calculations and the computational benefits obtained from the parallelization of the code are illustrated; in Section 4 results obtained for the title process and the rationalization of its dynamical features are discussed; in Section 5 some conclusions are drawn.

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2. Method

A full description of the quantum-classical method and of the code used has been reported elsewhere [8, 21] and here we outline only its application to the OH + H2 system. The considered quantities are the detailed (state

specific, for the reactive process, and state to state, for inelastic collisions)

40

properties for colliding diatom-diatom systems by solving the related time dependent Schr¨odinger equation as formulated below:

i¯h∂Ψ ∂t = ˆH

?Ψ, (1)

The nuclear arrangement is defined in a space fixed Jacobi coordinate system via the three vectors R1 (O-Hc), R2 (Ha-Hb) and R3 (OHc center

of mass-HaHb center of mass) (see Fig. 1). In this coordinate system the

Figure 1: Sketch of the fixed Jacobi coordinates used for the OHc + HaHb system

Hamiltonian can be expressed in terms of the cartesian coordinates of the three i-th r vectors (collectively indicated as {x, y, z}), obtained by mass scaling the R vectors:

ˆ H? = − 3 X i=1 ¯ h2 2µi  ∂2 ∂xi2 + ∂ 2 ∂yi2 + ∂ 2 ∂zi2  + V ({x, y, z}) (2) where µ1 and µ2 are the reduced mass of diatoms OHcand HaHb, respectively

and µ3 is the overall reduced mass of the system:

µ1 = mHcmO mHc+ mO , (3) µ2 = mHamHb mHa + mHb , (4)

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µ3 =

(mHa+ mHb)(mHc+ mO)

mHa+ mHb+ mHc+ mO

= (mHa+ mHb)(mHc + mO)

M (5)

Using the corresponding spherical polar representation xi, yi, zi can be

ex-pressed as:

xi = risin θicos φi (6)

yi = risin θisin φi

zi = ricos θi

where ri is the radial coordinate and θi and φi represent the associated polar

and azimuthal angles, respectively. In these coordinates Eq.2 can be refor-mulated as: ˆ H = − 3 X i=1 ¯ h2 2µi  ∂2 ∂ri2 + 1 ri2  ∂2 ∂θi2 + cot θi ∂ ∂θi + 1 sin2θi ∂2 ∂φi2  + V ({r, θ, φ}) (7) where {r, θ, φ} = r1, θ1, φ1, r2, θ2, φ2, r3, θ3, φ3 and ˆH = r1r2r3Hˆ?(r1r2r3)−1, 45

once that the wavefunction transformation Ψ = ψr1r2r3is adopted in order to

eliminate first derivatives in ri. The hallmark of the method is the already

mentioned different treatment of classical and quantal degrees of freedom (in the present case, vibrations are treated quantally while translations and rotations classically). For this reason, use has to be made of the semiclassical

50

Hamiltonian that, once replaced the momentum operators with their classical counterparts, takes the form (see eq. 6 of Ref. [21]):

ˆ HSC = ˆTQ+ 3 X i=1 1 2µir2i  p2θi+ 1 sin2θi p2φi  + p 2 r3 2µ3 + ˆV ({r, θ, φ}), (8) where ˆTQ represents the purely quantum kinetic energy operator defined as:

ˆ TQ= − 2 X i=1 ¯ h2 2µi ∂2 ∂r2 i . (9)

In order to couple the different degrees of freedom one has to consider the effective semiclassical Hamiltonian expectation value:

HSCef f = hψ| ˆHSC|ψi

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where the brackets denote integration over the variables r1 and r2. More

55

in detail, HSCef f can be written also as

HSCef f = hψ| ˆTQ|ψi hψ|ψi + p2 r3 2µ3 + 1 2µ3r32  p2θ3 + 1 sin2θ3 p2φ3  + 2 X i=1 hψ|1/r2 i|ψi hψ|ψi 1 2µi  p2θi + 1 sin2θi p2φi  +hψ| ˆV |ψi hψ|ψi . (11)

In order to follow the time evolution of the system, the time integration of classical degrees of freedom and the wavepacket propagation need to be performed simultaneously. For the latter one we can rewrite the time de-pendent Schr¨odinger equation by considering only the quantal part of the

60 Hamiltonian: i¯h∂ψ ∂t = ( 2 X i=1  − ¯h 2 2µi ∂2 ∂ri2 + 1 2µiri  p2θ i+ 1 sin2θi p2φi  + V (r1, r2; t) ) ψ. (12) and integrate it using a grid of internuclear distance values. To this end a two-dimensional discrete grid in the quantum variables r1 and r2 is adopted and

the initial wavefunction is formulated as the product of two Morse oscillator wavefunctions representing the vibrations of the two molecules,

65

ψ(r1, r2, t = 0) = ϕυ1(r1)ϕυ2(r2). (13)

The time propagation of the wavepacket is performed using the split operator. On the contrary, the classical variables are integrated using the predictor corrector method. The reactive probability is computed when a trajectory is completed by evaluating the wavefunction flux before its ab-sorption by an imaginary absorbing potential [22] located in the product

70

channel, while the non reactive one is obtained by analysing the remaining non reactive wavepacket. To this end the final wavefunction is projected onto the reactants basis functions to evaluate the final population distribution of the vibrational levels.

Note that in this case the evaluation of reactive and inelastic probabilities

75

is performed simultaneously. This is somewhat different from the mixed quantum classical method introduced by Billing [23] and recently extended

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by Babikov (see [24] and references therein) which is focused on inelastic scattering.

3. Computational Details

80

For the calculations to be reliable, convergency tests aimed at optimizing the simulation parameters have been performed. The parameters obtained from the optimization procedure are summarized in Table 1. As to the ex-tension of the internuclear distance grid, simulations at different upper and lower grid limits for both rOHc and rHaHb (r1 and r2 in the notation of the 85

equations of Section 2) have been performed. Tests have been carried out at 0.3, 0.4, 0.5 and 0.6 ˚A to set the minimum value of rOHc (rOHcmin) and

at 1.5, 2.0, 2.5, 3.0, 4.0 ˚A to set its maximum value (rOHcmax). Similarly

for rHaHbmin and rHaHbmax tests have been performed at rHaHb equal to 0.15,

0.20, 0.3, 0.4 and 0.5 ˚A and at 3.0, 4.0 and 5.0 ˚A, respectively. The initial

90

value of the diatoms centers of mass distance, r3, was also tested. However,

it seems that the potential energy surface is accurate up to r3= 8 ˚A ca., as

also suggested by the fact that fitting points in ref. [13] extend up to ca. 15 bohr. Thus, the value of r3= 7 ˚A was chosen as initial parameter for all

simulations.

95

Different ∆t values have also been used (0.01, 0.02, 0.04, 0.08, and 0.1 fs) to the end of evaluating the convergence of the treatment within the integration time step. The number of trajectories has also been optimized at the lowest energy of 0.1 eV by checking the convergence of the results. Table 1 shows the adopted values of the various parameters.

100 Parameter Value NrOHc 42 NrHaHb 81 rOHc min, rOHc max, ∆rOHc (˚A) 0.40 2.5 0.05 rHaHb min, rHaHb max, ∆rHaHb (˚A) 0.15 4.0 0.05 r3(˚A) 7.0 ∆t (fs) 0.04

Number of trajectories at each energy value 30,000

Table 1: Optimized parameters used for the simulations. NrOHc and NrHaHb represent the

number of grid points along rOHc and rHaHb coordinates, respectively; r3 is the distance

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In order to gain computing efficiency we have parallelized the Fortran code using the shared memory paradigm in its OpenMP implementation. The parallel distribution of the workload was performed by assigning dif-ferent trajectories to the various threads. As expected, this coarse grain parallelization leads to performances close to ideal (solid line) upon an

in-105

crease of the number of processors (see Figure 2 for the related speedup plot when increasing the number of processors up to 8).

Figure 2: Measured Speedup (crosses) of the QC code using up to 8 cores. The solid line shows the ideal behaviour.

4. Results and Discussion

State specific reactive QC probabilities were calculated for total angular momentum J = 0 and compared with the full quantum ones of Chen et

110

al. [13] at different initial translational energies (Figure 3). The Figure shows the good agreement of the two results which differ for less than 10%, with

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the QC probabilities slightly overestimating the full quantum ones in the vibrational ground state for the two diatoms.

Figure 3: State specific (J = L = 0, vOH = vH2 = 0, jOH = jH2 = 0) OH + H2 → H2O

+ H reactive probability plotted as a function of the translational energy (QC results in red, full quantum results in green)

A further qualitative analysis can be made by comparing QC results with

115

the experimental ones. The latter show a marked increase of the reaction rate constant when H2 (note that hereinafter we drop the labelling of

hydro-gen atoms in H2 and OH because we always refer to the arrangement of the

reactants.) is vibrationally excited [25] (specifically for vH2 = 1 with respect

to vH2 = 0), whereas no significant change in the reaction rate is observed 120

upon vibrational excitation of OH [26] (for vOH = 1 with respect to vOH = 0).

Indeed, our results confirm the higher efficiency of the initial vibrational exci-tation of the reactant H2 in promoting the reaction (see Figure 4). The plots

of the QC reactive probability (compare crosses (vOH = 0, vH2 = 0) with

points (vOH = 0, vH2 = 1) and solid circles (vOH = 0, vH2 = 2)) of Figure 125

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vH2, that is from 0.1 to 0.2 larger than that of the neighbour lower

vibra-tional state in the whole investigated collisional energy range. In contrast, the vibrational excitation of OH from vOH = 0 to vOH = 1 and vOH = 2

has no appreciable effect on the value of the reactive probability of the

sys-130

tem: as shown in Figure 4, the reactive probability computed for different values of the vibrational quantum numbers of OH are in this case hardly distinguishable from that of vOH = 0, in agreement with the experimental

results of ref. [26] and the theoretical ones of refs. [7, 27]. In particular, our results also show a very small increase in the reaction probabilities for

135

vOH = 1, with respect to vOH = 0, at small energies (not appreciable in Fig.

4). This difference in the reaction probabilities tends to vanish as the energy increases.

Figure 4: State specific (L = 0, jOH = jH2 = 0) OH + H2 → H2O + H QC reactive

probability plotted as a function of the translational energy at different initial vibrational levels of the reactants.

The good agreement between QC and full quantum approaches for vOH =

vH2 = 0 and, qualitatively, between QC results and experiments is a signif-140

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icant test for the present approach, taking into account that the computa-tional effort of these calculations is only slightly larger than quasi-classical methods and that relevant quantum effects (like the zero point energy of the diatoms) are included in the treatment.

Once proven that QC calculations can provide reliable results, we focused

145

our investigation on the understanding of the reaction mechanism. For this reason, we have analyzed the trajectories associated with the QC calcula-tions performed using the same initial condicalcula-tions of the quantum study. The product H2O molecule is formed by the binding to OH of one of the

hydro-gens in which H2 breaks. This is in agreement with the fact that, as shown

150

in Fig. 4, an increase of the vibrational excitation of the hydrogen molecule (and, therefore, the stretching of the related bond) is indeed the key factor in promoting reactivity.

Typically, the trajectories associated with the initial conditions of the reactants mimicking those of quantum calculations follow two qualitatively

155

different paths to reactions. The first one (see Fig. 5, upper panel) is con-sistent with a direct mechanisms in which H2 dissociates and one of the

dissociated H sticks on the OH molecule while the remaining hydrogen atom flies away. The second one (see Fig. 5, lower panel) is typical of a long lived intermediate complex forming mechanism that leads to more effective energy

160

redistributions among the various degrees of freedom.

Our calculations also allowed to single out an effect that only full quantum and QC calculations can spot, while it might go undetected in pure quasi-classical trajectory calculations due to their poor discretization method. This effect is the vibrational excitation of the non reacted OH molecule as a result

165

of the already commented formation of an intermediate long lived complex. This is clearly shown in Figure 5, lower panel, where the quantum expectation value of rOH after collision becomes sensibly larger.

The possibility of producing vibrationally excited OH after collision is confirmed by the calculations of the probabilities for vibrational energy

ex-170

change process. An accurate quantitative evaluation of vibration-to-vibration and vibration-to-translation energy transfer cross sections and rate constants would require an initial distance of the centres of mass of the diatoms much larger than 8 ˚A, which, as already mentioned, seems to be the largest dis-tance at which the PES is still reliable. Indeed the long range part of the

175

potential is known to play a crucial role in these non reactive events, par-ticularly when resonant or near resonant processes are concerned. Because of this, here we will only make qualitative considerations about the diatoms

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Figure 5: Plot of a direct (upper panel) and intermediate forming (lower panel) trajectories for J =0, vOH=0, vH2=0. Classical values of rOHa (dashed-dotted line), rOHb (double

dashed-dotted line) and quantum expectation values of {rHaHb} =

hψ|rHaHb|ψi

hψ|ψi (dotted

line) and {rOHc} =

hψ|rOHc|ψi

hψ|ψi (solid line) are reported as a function of propagation time.

vibrational population distribution after non reactive collisions.

Figure 6 shows the QC state to state non reactive probabilities

corre-180

sponding to selected initial vibrational states as a function of the product vibrational level at fixed reactant collisional energies. In every case vibra-tionally elastic collisions are largely predominant, and they have been re-moved from the graphs to better focus on vibrational energy redistribution. Panels referring to the initial (vH2 = 0, vOH = 0) state show that population 185

of the first excited vibrational states only occurs, with very small probability, when the initial translational energy is larger than ∆E = Ev=1− Ev=0, being

∆E = 0.516 eV for H2 and ∆E = 0.442 eV for OH.

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ex-change of one quantum of vibrational energy between the collision partners

190

is more likely for (vH2 = 1, vOH = 0) → (vH2 = 0, vOH = 1) than for

(vH2 = 0, vOH = 1) → (vH2 = 1, vOH = 0) process, and even more so for

(vH2 = 2, vOH = 0) → (vH2 = 1, vOH = 1) with respect to (vH2 = 0, vOH =

2) → (vH2 = 1, vOH = 1) process. This might be an indication of a

ten-dency to the depletion of excited H2 and an increase of excited OH in the

195

final states. Furthermore, at higher translational energies vibrationally ex-cited H2 is more prone to react and, thus, is much less efficient in populating

higher vibrational levels.

Figure 6: Non reactive state (vOH, vH2) to state (v

0 OH, v

0

H2) probabilities plotted as a

function of one of the product vibrational level at Etr=0.1 eV, solid line, Etr=0.3 eV,

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5. Conclusions and Future Perspectives

In this paper we use a Quantum Classical technique in order to perform

200

a detailed state to state study of the title process and to obviate the heavy computational demand associated with the use of full quantum approaches. Furthermore, we exploit the fact that we have been able to gain almost one order of magnitude in speed by running in parallel on a cluster of 8 processors. This has been made possible both by the largely decoupled nature of the

205

classical mechanics component of the QC code and by the compactness of its quantum one that makes the code highly distributable.

Our calculations led to an excellent agreement between full quantum and QC results. The QC calculations provide a clear evidence for the fact that there exist two different reactive mechanisms for the OH + H2 → H2O +

210

H reaction: the first one proceeding through a direct attack of the H atom originating from the dissociation of H2 and a second one forming a long

lived complex in which the two H atoms play a longer living interchange of energy among the different degrees of freedom. The two mechanisms also offer a rationale for understanding why the behavior of non reacted

215

H2 is mostly vibrationally adiabatic (the largest fraction of vibrationally

excited H2 ends up by reacting, as shown also by the fact that vibrational

excitation of H2 significantly increases reactivity), while the final vibrational

distribution of non reacted OH shows a tendency to give higher vibrational energy distributions.

220

Preliminary calculations on internal energy redistribution processes upon non reactive collisions have also demonstrated that the QC code can be effec-tively used to investigate quantum effects in inelastic scattering. However, in order to get reliable results in this sense, the potential energy surface has to take into account long range effects, which are essential to accurately

225

evaluate the exchange of vibrational quanta of energy.

To this end future studies, beside being extended to include larger col-lision energies and higher vibrational excitations and to calculate averaged quantities so to allow the computation of the relevant rate constants, will be directed on the investigation of how the nature of the potential, and

partic-230

ularly the introduction of a long range tail to the present PES, affects both the reactive and inelastic energy exchange processes.

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Acknowledgements

Prof. D.H. Zhang and Dr. J. Chen are gratefully acknowledged for provid-ing the potential energy surface. Financial support is acknowledged from the

235

Phys4entry FP7/2007-2013 project (Contract 242311) and the EGI-Inspire project (Contract 261323). The European Grid Infrastructure (EGI) through the COMPCHEM Virtual Organization, the Italian CINECA computing cen-tre and the Supercomputing Center for Education & Research (OSCER) at the University of Oklahoma (OU) are acknowledged for providing

comput-240

ing resources and services. CM thanks The European Joint Doctorate on TCCM project ITN-EJD-642294-Theoretical Chemistry and Computational Modeling. Partial financial support by the Fondazione Cassa Risparmio di Perugia (Project Code 2015.0331.021 Scientific and Technological Research) is gratefully acknowledged.

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Figure

Figure 1: Sketch of the fixed Jacobi coordinates used for the OH c + H a H b system
Table 1: Optimized parameters used for the simulations. N r OHc and N r HaHb represent the number of grid points along r OH c and r H a H b coordinates, respectively; r 3 is the distance between the centers of mass of the diatoms.
Figure 2: Measured Speedup (crosses) of the QC code using up to 8 cores. The solid line shows the ideal behaviour.
Figure 3: State specific (J = L = 0, v OH = v H 2 = 0, j OH = j H 2 = 0) OH + H 2 → H 2 O + H reactive probability plotted as a function of the translational energy (QC results in red, full quantum results in green)
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