Quantum Dynamics of the 18O + 36O2 Collision Process - The Journal

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Quantum Dynamics of the O + O Collision Process Gregoire Guillon, Tammineni Rajagopala Rao, Susanta Mahapatra, and Pascal Honvault J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b08163 • Publication Date (Web): 05 Oct 2015 Downloaded from http://pubs.acs.org on October 8, 2015

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The Journal of Physical Chemistry

Quantum Dynamics of the

18

36

O+

O2

Collision Process Grégoire Guillon,† Tammineni Rajagopala Rao,†,¶ Susanta Mahapatra,∗,†,‡ and Pascal Honvault∗, †Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université de Bourgogne Franche-Comté 21078 Dijon Cedex, France, UFR Sciences et Techniques, Univ. de Franche-Comté, 25030 Besancon cedex, France ‡School of Chemistry, University of Hyderabad, Hyderabad 500046, India ¶Current address: School of Chemistry, University of Hyderabad, Hyderabad 500046, India, Germany E-mail: [email protected]; [email protected] Abstract We report full quantum cross sections and rate constants for the 36 O

2

+

18 O

18 O

+

36 O

2

−→

collision process. This constitutes to the best of our knowledge the first

dynamical study of the

18 O18 O18 O

system, with three identical

emphasize the comparison with the

16 O

+

32 O

2

18 O

oxygen atoms. We

collision as this latter presents the

exact same features as the one treated here, except the consistent change of mass for all three atoms. We find very similar behaviors in the cross sections and we confirm that the rates are faster when three identical nuclei are involved. In particular, we cannot dynamically study this system with classical trajectory methods and we have to include properly the indistinguishability of the three differences with the

16 O16 O16 O

18 O

nuclei. However, we note some slight

benchmark system, and we focus our analysis on their

origin.

1

ACS Paragon Plus Environment


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1. Introduction In a recent paper, 1 we have presented full quantum cross sections and rate constants for the collision involving only the most abundant isotope of oxygen 32

O2 −→

step

16

32

O+

O2 + 32

16

16

O, namely, the

16

O +

O process. At low pressure, this reaction is believed to be the initiation

O2 −→

48

O∗3 for ozone formation, leading to a highly rovibrationally excited

ozone complex O∗3 before it decays back into O + O2 fragments. The efficiency of the O + O2 collision process is important as it strongly competes with the subsequent deactivation process of O∗3 which might take place in presence of a third body M (most likely molecular nitrogen N2 , very abundant in the whole atmosphere) carrying out the excess energy to eventually give stable ozone: O∗3 + M −→ O3 + M. We have highlighted in particular a strong quantum effect due to the indistinguishability of the three nuclei, leading to a rate constant more than one order of magnitude larger than when ignoring this effect. The efficiency of this process, which has been ignored thus far, is likely to deeply modify our understanding of ozone formation. As it happens, stratospheric ozone presents a significant anomalous enrichment in 17 O and 18

O isotopes, roughly equal for both of them, and thus called mass-independent fractionation

(MIF) effect. 2–4 Many attempts have been devoted to understand this puzzling phenomenon. It is now well accepted that the large isotope effect observed in the recombination reaction, O + O2 + M −→ O3 + M is the key mechanism responsible for the MIF. This effect is related to differences in lifetimes for variously substituted x Oy Oz O complexes (where x, y and z can stand for 16, 17 or 18) due to differences in zero-point energies (the so-called ∆ZPE) between two O2 fragments into which metastable ozone can dissociate. 5 Other factors have been evoked, like the η-effect. 6 However, this mass-independent fractionation effect is far from being well understood, for one simple reason: presently, despite a few attempts using classical-quantum hybrid methods (see for example Ref. 7 and references within), nobody is able to compute the rate for the recombination reaction O + O2 + M −→ O3 + M using full quantum methods. 2


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Concerning the exchange processes

18

32

O+

O2 and

16

O+

36

O2 alone (not even talking

of the recombination reaction leading to ozone), significant differences already appear between experimentally measured 8–11 and theoretically computed rate constants. 12–15 It is also important to note at this stage that there is no experimental data for the 18

O+

36

16

O+

32

O2 and

O2 collisions.

Many experimental data gathered from the literature of the last thirty years 2–4,6,16–23 tend to indicate that, apart from the reference

16

O16 O16 O species, asymmetric isotopically

substituted species seem to live longer than symmetric ones. Thus, 18 O18 O18 O and 17 O17 O17 O are rapidly depleted. In the case of

18

O18 O18 O, this due both to the rareness of

18

O (its

abundance is about 0.2 % in the atmosphere) and to the effectiveness of the associated exchange reaction The

17

18

O18 O +

18

O, as we will show in the following analysis of our results.

O17 O17 O system has extremely low abundance in the atmosphere, and is more exotic

and complicated because it involves several different nuclear spin functions. It deserves a study on its own right, less motivated by its atmospherical interest than by its peculiar properties at very low temperatures, of use in the cold chemistry community. This study is currently under progress in our group and will be presented in a future paper. The quantum dynamics calculations we performed 1 for the 16

16

O +

32

O2 −→

32

O2 +

O process, as well as other wave packet dynamical studies from other groups, 13–15 were

supported by a recently developed global potential energy surface (PES), 24 which we will call in the following sections the DLLJG PES, after the names of the authors, whose accuracy seems to be established by different dynamical studies, despite some remaining discrepancies with experiments. 12–15 In particular, the negative temperature dependence of the ratio of thermal exchange rates for

18

O+

32

O2 and

16

O+

36

O2 processes, k8/6 , is reproduced using

this PES, by different quantum methods. 12,14 This constitutes a good indication that the DLLJG PES is of reasonable accuracy for collision dynamics purposes, and we used it in the present whole study of the

18

O18 O18 O system.

The goal of this paper consists in obtaining completely new cross sections and rates for

3



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the

18

O18 O +

18

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O collision, taking the quantum symmtery of the problem explicitely into

account, comparing them to those recently obtained for the

16

O16 O +

16

O collision.

In section 2, we describe briefly the underlying time-independent quantum mechanical (TIQM) method used to obtain dynamical observables. In section 3, we discuss and analyse our results concerning the 16

18

O18 O18 O system, and we make various comparisons with its

O16 O16 O counterpart.

2. Theory We set out to obtain dynamical observables like cross sections and rate constants for the 18

O +

18

O18 O (from now on, we shall say 8 + 88) collision, at temperatures relevant to

atmospheric conditions, that is, between 220 and 330 K. However, for completeness, we will present rate constants for temperatures ranging from 200 to 500 K. This implies computation of cross sections for collision energies between 0.001 and 0.4 eV. For that purpose, we use a TIQM approach, which consists of solving coupled equations arising from the time-independent Schrödinger equation (TISE) expressed in democratic hyperspherical coordinates. The interested reader can find a detailed description of this TIQM approach in a number of papers. 25–28 A comprehensive review of the method applied to the O + O2 collision can also be found is our previous work 12 and the Supporting Information of Ref. 1 Briefly, within theses coordinates, one of them, the hyper-radius ρ, characterizes the size of the system whereas the two remaining ones are angles giving its shape. First of all, ρ-dependent adiabatic basis functions are computed. To this end, the domain of ρ is partitionned into 140 sectors between 3.25 and 14.30 a0 . The basis functions are obtained at the mid-point of each sector by solving the two-dimensional hyper-surface Schrödinger equation. The maximum value of the z-projection of the total angular momentum on the axis of least inertia of the complex, Ω, used in this calculation is critical in its computational cost. Here, we found that a value of Ωmax = 40 was enough to yield converged results. Then,

4


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for a given total angular momentum J, we expand the solution of the TISE in terms of the adiabatic states previously obtained. We have used 100 adiabatic states at J = 0, and the resulting coupled equations are solved using the log-derivative propagator. 29,30 The K and T-matrices are obtained at a large value of ρ by matching of the propagated solution to its asymptotic form, respecting the asymptotic boundary conditions for the scattering problem. We have obtained K-matrices at 240 energy grid points, using a maximum total angular momentum of Jmax = 190. Finally, the cross sections were obtained straightforwardly from the T-matrix elements, because we are interested in both nonelastic and elastic processes. The important point is that the 8 + 88 collision requires a proper treatment of the indistinguishability of the three function for the whole spin (as

16

18

18

O nuclei and the inclusion of the zero-nuclear spin

O18 O18 O (888) system in the dynamics. As

O), we are in the exact same situation as for the

16

18

O has zero nuclear

O16 O16 O (666) system. We

refer the reader to the Supporting Information of Ref. 1 for more details. In comparison with the 666 system, the 888 system presents only a mere mass scaling. We draw the reader attention on the fact that it is physically impossible to distinguish between what we call classically "reactive" process (with bond breaking and new bond formation) and pure inelastic process (with change of the diatom state but no classical rearrangement). Therefore, we will employ the same terminology as in our previous paper on the 6 + 66 collision. Namely, ’elastic’ means an energetically elastic process (with or without atom exchange) whereas nonelastic refers to all other processes.

3. Results and discussion 1 shows the 8 + 88 elastic (dashed red line), nonelastic (dotted red line) and total (elastic + nonelastic: solid red line) integral cross sections (ICSs) for the first initial rotational state j = 1 of O2 as a function of the collision energy, Ec , in comparison with its counterparts (respectively the dashed, dotted and solid black lines) for the 6 + 66 collision. For complete-

5



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Page 6 of 15

ness, we have plotted all the ICSs for a collision energy ranging from 0.001 to 0.4 eV. The 8 + 88 and 6 + 66 collisions follow the same pattern. Similarly to what happens for the 6 + 66 system, the elastic ICS is much larger than the nonelastic one. Actually, the overall shape of the total process is consistently following the shape of the elastic ICS, due to the high difference in magnitude between the elastic and nonelastic transitions. However, we may note some particular features for the 8 + 88 collision at low energies, and significant differences in magnitude at higher energies. The 8 + 88 elastic ICS strongly oscillates at low collision energies, as does the 6 + 66 ICS. These oscillations come from quantum resonances, closely associated with quasibound states supported by a 1.13 eV deep potential well in the O3 PES, giving rise to a relatively long-lived intermediate complex. As a consequence of a higher global mass for the 888 system, these quasibound states are shifted and yield slightly different energy positions for the peaks in the resonance structures of the 8 + 88 collision, not clearly visible in 1. These oscillating structures deserve a detailed study of its own right at collision energies close to zero and their analysis will be addressed in a future dedicated work. We note that both elastic and nonelastic ICSs monotonically decrease with increasing Ec . This is due to the absence of a barrier in the DLLJG PES for the entrance arrangment of the O + O2 fragments. We further remark that at higher collision energies, both 8 + 88 elastic and nonelastic ICSs decrease faster than their counterparts for the 6 + 66 collision. This is related to the higher mass for the whole 8 + 88 system, which makes the dynamics less reactive and a bit damped. Finally, we notice the existence of a threshold for the nonelastic transitions, for both the 8 + 88 and 6 + 66 systems. Indeed, the lowest populated rotational level for

36

O2

and 32 O2 is j = 1, and the lowest level accessible for a nonelastic process is j = 3, so there is an energy threshold of 10Be in the cross sections, Be being the rotational constant of or

32

O2 (whose value for

32

36

O2

O2 is roughly 1.44 cm−1 , 31 that is 1.78 10−4 eV) for a vibrational

level v = 0 of the 3 Σ− g electronic state. This threshold is very slightly lower for 8 + 88, due to the mass difference, but the difference is too small to appear clearly in 1.

6


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1400

8+88 total 8+88 elastic 8+88 nonelastic 6+66 total 6+66 elastic 6+66 nonelastic

0

2

)

1200

Integral cross section (a

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


1000 800 600 400 200 0

0.0

0.1

0.2

0.3

0.4

Collision energy (eV)

Figure 1: Comparison of the total, elastic and nonelastic integral cross sections (in the unit of a20 ) for the first initial rotational state of O2 (v = 0), j = 1, obtained in the 8 + 88 and 6 + 66 collisions, as a function of collision energy in eV. We show the corresponding rate constants in 2, respectively associated with the total, elastic and nonelastic ICSs, where the line style follows exactly the pattern used for the plotting of cross sections in 1. Their behaviours are very smooth and monotoneous, all increasing with temperature. For both 888 and 666 systems, the elastic component is rising faster than the nonelastic one, giving a significant increase of the total rate for higher temperatures. When comparing the 8 + 88 and 6 + 66 collisions, all three kinds of rates (total, elastic and nonelastic) are very close side by side. Also, in agreement with the ICSs analysis, rates associated with the 8 + 88 collision remain slightly lower than their 6 + 66 counterparts, for nearly all temperatures. Accordingly, the seemingly tiny differences in resonance structures in ICSs at low Ec and the soft drop of the 8 + 88 total ICS at higher Ec are clearly apparent in the rate constants. In 3, we present the evolution of the nonelastic integral cross sections as a function of the

7



1.8e-9 8+88 total 8+88 elastic 8+88 nonelastic 6+66 total 6+66 elastic 6+66 nonelastic

1.6e-9 1.4e-9

Rate constant (cm 3.s-1)

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

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1.2e-9 1.0e-9 8.0e-10 6.0e-10 4.0e-10 2.0e-10 0.0 100

200

300

400

500

Temperature (K)

Figure 2: Comparison of the total, elastic and nonelastic rate constants (in the unit of cm3 s−1 ) for the first initial rotational state of O2 (v = 0), j = 1, computed for the 8 + 88 and 6 + 66 collisions, as a function of temperature in K. final rotational state of O2 , which can take only even values due to nuclear spin statistics, for both the 8 + 88 and 6 + 66 systems, at four distinct collision energies. The first striking point we notice is that, except for the lowest collision energy presented here, the contibutions to the cross section for each odd value of j ′ is nearly exactly the same for both systems. This fact highlights the very strong similarity between them. We note that for the two lowest collision energies, namely Ec = 0.005 eV and Ec = 0.01 eV, the distributions are monotonically decreasing, the highest contribution coming from the lowest rotational level, j ′ = 3. In these two first cases, the contribution to the ICS is larger for the lighter 666 system, and also decreases faster. This reflects the fact that low j nonelastic processes are favoured at low collision energies. The lighter the system, the more pronounced is this tendency. Now, for the two higher Ec , 0.05 and 0.1 eV, the monotonic character of the distribution, which is broader, is lost and we have a maximum at j ′ = 5. Products are allowed to be formed in some higher rotationally excited states, whatever the mass of the 8


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100

150 0.010 eV

6+66 8+88

100

75

vʼ = 0

2

ICS (a0 )

0.005 eV

50 50 25

0

0 3

5

7

3

5

7

9

40

25 0.100 eV

0.050 eV

20

30 2

ICS (a0 )

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


15 20 10 10

5

0

0 3

5

7

9

11 jʼ

13

15

17

19

3

7

11

15

19

23

jʼ

Figure 3: Selected product rotational state j ′ resolved ICSs for nonelastic processes at four well separated collision energies for v ′ = 0, for both the 8 + 88 and 6 + 66 collisions. system. Finally, we show in 4 the total angular momentum degenaracy-averaged reaction probabilities, or weighted opacity functions, as a function of the total angular momentum J, for both 8 + 88 and 6 + 66 collisions, at the same previously mentionned collision energies. These quantities are computed for a final vibrational state v ′ = 0 and summed over all j ′ , except j ′ = 1. They are therefore only related to nonelastic processes. We see that at the two lowest Ec , the probabilities are highly structured for both systems. In sharp contrast to that, their behaviour becomes very smooth and regular, with a neatly apparent maximum, starting from Ec = 0.05 eV. This maximum always occurs at a slightly higher value of J for the 8 + 88 collision, due to the larger mass of the latter system. The important point here is the striking similarity between the profiles, which reflects the 9



50

40 6+66 8+88 (2J+1)P(J)

30

0.005 eV

0.010 eV 40

vʼ=0, ∑ jʼ 30

20 20 10

10

0

0 0

5

10

15

20

25

30

35

0

10

20

30

40 200

100

0.100 eV

0.050 eV

150 (2J+1)P(J)

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

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75 100

50

50

25 0

0 0

20

40

60

80

0

20

40

J

60

80

100

J

Figure 4: Weighted reaction probabilities (opacity functions) as a function of total angular momentum J for nonelastic processes at four well separated collision energies for v ′ = 0, for both the 8 + 88 and 6 + 66 collisions. closeness of behaviour of the two systems on the reaction dynamical point of view.

4. Conclusions We have performed the first full quantum study of the 18 O + 18 O18 O collision taking account of the full quantum symmetry of the problem. The dynamical calculations are supported by a recently developed accurate potential energy surface known as the DLLJG PES. Cross sections for this collision have been obtained for a wide range of collision energies relevant to atmospheric conditions and beyond, and present very similar behaviors as the parent 16

O+

16

O16 O collision, 1,13 the 8 + 88 system being similar from the interaction potential,

indistinguishability and nuclear spin point of view, the only difference being a sole change 10


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in mass of all three atoms. As for this latter system, we find that the total ICSs are very large for the 8 + 88 collision, comparable to those for the 6 + 66 collision. This fact is due to the same huge quantum symmetry effect observed for the 666 system. The work reported here may help to understand the very fast depletion of

18

O18 O18 O observed in the upper

atmosphere.

Acknowledgments TRR thanks Department of Science and Technology, New Delhi for the RA fellowship (Project No. SB/S1/PC-052/2013). TIQM computations have been done on the cluster of the Centre de Calcul de l’Université de Bourgogne (CCUB, Dijon, France). We thank Richard Dawes for sending us the O3 potential energy surface.

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O2 isotope exchange reactions: Dominant role of reactive resonances

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1400

18

O

O + O2

0

2

)

1200

Integral cross section (a

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1000 800 TOTAL

600 400

ELASTIC

200 NON ELASTIC

0

0.00

0.02

0.04

0.06

Collision energy (eV)

15


0.08

0.10

Quantum Dynamics of the 18O + 36O2 Collision Process - The Journal

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