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Kinetics of the H + CH # CH + H Reaction At Low Temperature Ernesto Garcia, Pablo G. Jambrina, and Antonio Lagana J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b06212 • Publication Date (Web): 02 Aug 2019 Downloaded from pubs.acs.org on August 6, 2019

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The Journal of Physical Chemistry

Kineti s Of The H + CH → CH + H Rea tion At Low Temperature 2

∗ ,†

Ernesto Gar ia,

Pablo G. Jambrina,

‡

2

¶,§

and Antonio Laganà

Departamento de Quími a Físi a, Universidad del País Vas o (UPV/EHU), Paseo de la Universidad 7, 01006 Vitoria, Spain, Departamento de Quími a Físi a, Universidad de Salaman a, Plaza de los Caidos, 37008 Salaman a, Spain, UOS Perugia, CNR ISTM, via El e di sotto 8, I-06123 Perugia, Italy, and Master UP srl, Strada Sperandio 15, I-06125 Perugia, Italy E-mail: e.gar iaehu.es

∗

To whom orresponden e should be addressed

†

Universidad del País Vas o (UPV/EHU)

‡

Universidad de Salaman a

¶

CNR ISTM

§

Master UP

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 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

Abstra t A quasi lassi al traje tory study of the kineti s of the title astro hemi al rea tion in the range of temperature varying from 5 K to 1000 K ( orresponding to both the outer and the inner regions of protostar and the ir umstellar envelops) was arried out and a lear dependen e of the rate oe ient on the temperature is given, in ontrast with the onstant value adopted in kineti s astro hemi al databases. Levering the massive nature of the performed al ulations and the detailed dynami al investigation of the rea tive pro ess, a rationalization of the temperature dependen e of the released translational energy as well as of the rovibrational population of the CH and H2 diatomi produ ts is also provided. Furthermore, the ee t of the initial rovibrational energy of CH2 on the state-spe i rate oe ients and ross se tions is analysed in order to single out the role played by the dierent regions of the potential energy surfa e on the dynami al out omes and on the modelling the temperature dependen e of the rea tive e ien y of the investigated pro ess. This led to a parametrization of the

omputed rate in terms of the following double Arrhenius expression (in cm3 s−1 ) k(T ) = 2.50 × 10−10 exp(−1.67/T ) + 5.98 × 10−11 exp(−280.5/T ) alternative to the

pie ewise formulation into the three subintervals of temperature in whi h the overall 5 − 1000 K interval an be divided.

2


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1

INTRODUCTION

Hydrogen atom (H), hydrogen mole ule (H2 ), methylidyne (CH), methylene (CH2 ) and methyl (CH3 ) radi als are hemi al spe ies of relevan e in ombustion pro esses and in astro hemistry. A

ordingly, in order to properly model rea tions involving them, a

urate estimates of their rate oe ients need to be provided to the relevant kineti s databases. However, a

urately determined low temperature (it is ommonly a

epted that the temperature T in the interstellar medium (ISM) is of the order of 10 K) values of rea tion rates are seldom available in the databases with the ex eption of those obtained from measurements performed at the CRESU (Fren h a ronym for "Kineti s of Rea tions in Uniform Supersoni Flows") fa ility. As a matter of fa t, CRESU experiments provided a

urate estimates of the rate oe ients for the CH + H2 → H + CH2 rea tion in the T range 13 K - 744 K 1 and for the rea tions of the methylidyne radi al with CH4 , C2 H2 , C2 H4 , C2 H6 , and C4 H8 in the T range 23 K - 295 K. 2 Unfortunately, no low temperature measurements are yet available for the title rea tion. The only available experimental values are those for the 298 K - 3000 K 3 interval of temperature. In addition, reported values dier signi antly from those for T = 300 K ranging from

8.30 × 10−11 to 4.92 × 10−10 cm3 s−1 . Dis repan ies persisted when the experimental values were reviewed for in orporation into high temperature kineti databases. As a matter of fa t, while Warnatz 4 opted for the onstant value of 6.64 × 10−11 cm3 s−1 Tsang and Hampson 5 opted for the other onstant value of 2.71 × 10−10 cm3 s−1 , (for the T range 300 − 2500 K), Baul h et al. 6 re ommended for the range 300−3000 K the expression 1.0×10−11 exp[900/T ] whi h leads to a value of 2.01 × 10−10 cm3 s−1 at 300 K and to de reasing values at higher temperature. The most re ent revision of Baul h et

al. 7 of kineti data for ombustion mod-

elling re ommended for the range 298 − 3000 K a onstant value of 2.0 × 10−10 cm3 s−1 . This is the value reported in the astro hemistry spe i databases like the KIneti Database for Astro hemistry KIDA 8 and the UMIST (University of Man hester Institute of S ien e and Te hnology) Database for Astro hemistry UDfA 9 for the temperature ranges of 10 3



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298 K and 10 - 2500 K, respe tively. On the ontrary, in the OSU (Ohio State University) database 10 the value of 2.7 × 10−10 cm3 s−1 is given for the 10 − 280 K range of T . The latter value is also the one quoted in refs 11 and 12 on the ground of experiments performed at room temperature while the former value is, as re ommended by I.W.M. Smith, 8 an average between the previous values and the one obtained in ref 13 at the same temperature. Out of their experiment 14 (performed at temperatures ranging from 185 and 800 K) Fulle and Hippler inferred, instead, a very mild positive temperature dependen e (2.2 × 10−10 (T /300)0.32 ) with su h expression being the same as the one used by Hébrard et al. in their photo hemi al model of Titan's atmosphere. 15 On the theoreti al side, the only ab initio based potential energy surfa e (PES) proposed for the rea tion

˜ 3 B1 ) → H2 (X1 Σ+ ) + CH(2 Π) H(2 S) + CH2 (X g

(1)

and its reverse is the MHG 16 PES developed in the year 2006. The MGH PES suggests that rea tion (1) is exoergi by 4.1 k al mol−1 (4.6 k al mol−1 when in luding zero-point energies), has no barrier and involves a deep well of 115.4 k al mol−1 asso iated with the

˜ 2 A′′ ). The kineti s and dynami s of the reverse formation of the methyl radi al, CH3 (X 2 rea tion (CH + H2 → H + CH2 ) were investigated on the MGH PES using both quantum and

lassi al te hniques. 1720 Quantum me hani al al ulations, assuming the omplex-forming mi ros opi me hanism of the rea tion, estimate rst the probability of forming the CH3

omplex using a time-dependent wave-pa ket treatment and then applying the phase spa e theory to obtain the rea tion rate oe ient. 17,18 Due to its relatively high endoergi ity, the CH + H2 → H + CH2 rea tion annot take pla e at low temperature. Its astro hemi al interest is, however, the radiative asso iation of CH and H2 to form CH3 in interstellar

louds. 21 The present work is part of the eort of the Mole ular S ien e ommunity to ontribute to the European Open S ien e Cloud (EOSC) Pilot proje t 22 by implementing MOSEX (Mole ular S ien e Enabled Cloud Servi es) 23 to develop a virtual environment for produ 4


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ing, storing, managing, analysing and re-use data for resear h, innovation and edu ation purposes. MOSEX is the out ome of the eorts made by the Mole ular S ien e (MS) ommunity within the D23 (METACHEM) 24 and D37 25 (GRIDCHEM) COST 26 A tions and within the EGEE-III 27 and EGI Inspire 28 european initiatives through the COMPCHEM Virtual Organization 29 and the CMMST Virtual Resear h Community 30 that is now being in orporated into the EOSC 31 Pillar 32 Proje t started on July 1st 2019. It leverages an OpenSta k loud image (VHERLA) 23 implemented at the GARR Cloud node of Palermo (Italy) as a Virtual Data Center. VHerla servi es are aimed to produ e/dis over data set(s) metadata querying, a

ess/download data set(s), run/ he k/terminate experiment/ omputations using downloaded data, intermediate data, in orre t/useless runs, preserve: orre t results for further use, support reprodu ibility of extremely large and streaming data when input data set(s) annot be stored, annotate, urate, preserve produ ed data by y ling from raw measured data, to annotated experimental data and metadata for a better re-use, to simulation based formulation of me hanism and observables for onsisten y he ks. The

y le is repeated to a satisfa tory onvergen e for validated availability in loud. MOSEX, operates at present on: (1) elettroni stru ture and equilibrium mole ular onguration properties for spe tros opy, synthesis, photovoltai and photo hemi al pro esses studies; (2)

lassi al and quantal rea tive and non rea tive e ien y studies for elementary and omplex detailed kineti s; (3) stru ture-property relationships studies for pharma ologi al and biologi al systems; (4) management of some distributed repositories and databases for hemi al systems; (5) dissemination, publi ation and assessment of mole ular knowledge via the R e-tests, the VIRT&L-COMM e-magazine and G_LOREP Learning obje ts disEChemtest

tributed repository. Aim of the present work along the above the mentioned MOSEX a tivities is to study for the rst time the low temperature kineti s and dynami s of rea tion (1) so as to obtain information of interest for the astro hemi al databases while gaining insight into its detailed me hanism. The paper is stru tured as follows. In Se tion 2 the used theoreti al methods 5



are des ribed, mainly the appli ation of the quasi lassi al traje tory (QCT) methodology to the title atom-triatom rea tion. In Se tion 3 the dependen e on the temperature of the rate

oe ient, the partitioning of the energy released when rea ting and the ee t of the CH2 rotation on the e ien y of the rea tive pro ess are dis ussed. In Se tion 4 the dependen e of the ross se tion on the ollision energy for several rovibrational states and the underlying mi ros opi me hanisms are analysed. In Se tion 5 the exible pie ewise parametrizations of the rate oe ients for in lusion in kineti s databases are provided. In Se tion 6 some nal on lusions are drawn.

2

THEORETICAL METHODS

QCT methodology has demonstrated to be suitable to investigate the kineti s and dynami s of omplex-forming rea tions of astro hemi al interest (see, for example, refs 3353). Re ently, some attempts to investigate the omplex-forming rea tion using the ring polymer mole ular dynami s methodology have been made. 43,5457 QCT al ulations were performed using a version of VENUS96 58 modied to in orporate the MHG PES, with derivatives being al ulated numeri ally. Another modi ation of VENUS96 was implemented to improve the sampling of the rovibrational states of the CH2 triatom. The original version of the ode sele ts separately for ea h traje tory the vibrational and the rotational energy. The vibrational energy is al ulated using a normal mode analysis and then it is sampled from a Boltzmann distribution at a given temperature. The rotational energy and angular momentum (J ) for CH2 are hosen from a thermal distribution by assuming a lassi al-me hani s symmetri top mole ular geometry. 59 As a result, both the rotational energy and the angular momentum exhibit ontinuous distributions with a null probability for J = 0. In order to avoid this limitation (typi al of lassi al me hani s treatments), we modied the ode by introdu ing a model quantum-me hani al formulation of the angular momentum 6


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and rotational energy. A

ordingly, even if the rigid CH2 triatom is an asymmetri top system having the following values of the rotational onstants: a = 54.75, b = 8.46 and

c = 7.33 m−1, we avoided the ompli a y of the related formulation 60,61 by leveraging the fa t that the rigid CH2 triatom is well approximated by the limiting prolate symmetri top

(b ≈ c) formula. Its asymmetry parameter κ, dened as κ = (2b − a − c)/(a − c), has, in fa t, a value of −0.952 (very lose to −1 the asymmetry value of a prolate symmetri top mole ule). A

ordingly, the expression for its rotational energy redu es to: 62

1 E(J, K) = (b + c) J(J + 1) + 2

1 a − (b + c) K 2 2

(2)

with J and K being the rotational quantum numbers (K ≤ J ) and K > 0 states being doubly degenerate. The al ulated rotational energy using equation 2 agrees, within a 2% error, with the exa t value for the lower rotational states. The rotational energy for ea h (J, K ) rotational state is then added to the normal mode vibrational energy for ea h vibrational state and then the sampling is arried out from the thermal distribution of the rovibrational energies. It has to be noted here also that, when starting a traje tory the a tual initial values of

J and K may dier slightly from the sele ted ones. In fa t, in both the standard and our version of the ode the sele tion of atomi oordinates and momenta depends only on both vibrational energy and the displa ement from equilibrium of the internu lear distan e. This introdu es a spurious ontribution to the value of the rotational angular momentum that

an be ompensated via a s aling of the internu lear distan es to the end of reprodu ing the total internal energy (that is a sum of vibrational and rotational energy when separability of both motions is assumed). As an example of the dierent distributions al ulated with the original and the modied version of the ode, Figure 1 shows the orresponding urves (normalized to the maximum value) at T = 10 and 50 K. As apparent from the Figure, the dis ontinuous quantum weight of J = 0 is signi antly dierent from the lassi al one with

7



the dis repan y de reasing as temperature in reases. The dieren es between lassi al and quantum values are smaller for J > 0 and there is also a dependen e on K (though not shown in the Figure). The parameters used to integrate the traje tories dier from those of refs 19,20 for the study of the reverse pro ess (see rea tion 1) due to the lower temperatures/energies

onsidered here and to the need of a hieving higher a

ura y. In parti ular, the time step was set to 0.01 fs (down from the previous value of 0.05 fs) in order to guarantee a onservation of total energy typi ally better than 5 × 10−4 k al mol−1. The total angular momentum is even better onserved. Initial and nal separations of the ollision partners were set equal to 25 Å (up from the previous value of 12 Å) in order to ensure that the intera tion energy between the fragments is negligible with respe t to the available one so as to allow the initial fragments experien e the true long-range intera tion, whi h is spe ially relevant at low

ollision energy. As a matter of fa t, the attra tive long-range intera tion leads to a value of the maximum impa t parameter to rea tion as large as bmax = 13.7 Å at T = 5 K. The value of the maximum impa t parameter de reases gradually as the temperature in reases, with

bmax = 8.7 Å being the value hosen at T = 1000 K. At a given temperature, the impa t parameter for ea h traje tory was hosen randomly a

ording to the fun tion bmax R1/2 , with

R being a random number. All remaining parameters needed to dene the initial onditions of ea h traje tory (vibrational phases and spatial orientation of mole ules) were sele ted randomly. The ollision between the H atom and the CH2 radi al an lead both to ex hange and abstra tion rea tive events. In ex hange rea tions one of the H atoms initially bound to C is repla ed by the olliding H atom. On the ontrary, in abstra tion rea tions an H atom produ ed by the breaking of a C-H bond of CH2 forms H2 with the olliding H atom. The breaking of both C-H bonds of CH2 , the formation of a bond between these two H atoms and the formation of a new bond between C and the olliding H atom is also onsidered an abstra tion rea tion. Indeed, the probability of this event is the same as that of the 8


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abstra tion of an hydrogen atom and the formation of a H2 mole ule. The hara terization of abstra tion rea tions is ompleted by the al ulation of the quantum-like rovibrational states of the nal diatoms. The diatomi rotational quantum number is al ulated using the orresponden e prin iple J ′ = − 12 + 21 (1 + 4J~ ′ J~ ′ /~2 )1/2 , being J~ ′ the rotational angular momentum, while the diatomi vibrational quantum number is worked out by semi lassi ally dis retizing the a tion integral of the vibrational motion. 63 The ontinuous integer rotational and vibrational quantum numbers are then rounded to the nearest integer value. The QCT thermal rate oe ient k(T ) is estimated as:

k(T ) = gel

8kB T πµ

1/2

π b2max

Nr N

(3)

where kB and µ are the Boltzmann's onstant and the redu ed mass of the rea tants, respe tively, bmax is the maximum value of the impa t parameter leading to rea tion and Nr is the number of rea tive events out of the total number of integrated traje tories N generated by sampling both the ollision energy and the initial rovibrational states of the triatom a

ording to the Boltzmann distribution orresponding to the given T value. The orresponding one standard deviation of the estimation is roughly proportional to (Nr )−1/2 . Note that eq 3 in ludes an ele troni fa tor, gel = 1/3, taking into a

ount that rea tants orrelate with doublet and quartet states and produ ts with only doublet states. 8 In the present paper, in order to rationalize the full thermal rate oe ient k(T ), we

al ulated the rovibrational state-spe i ross se tions σ(Etr , v, J, K) using:

σ(Etr , v, J, K) = π b2max

Nr (Etr , v, J, K) N(Etr , v, J, K)

(4)

where Nr (Etr , v, J, K) is the rea tive subset of the total number of integrated traje tories

N(Etr , v, J, K) generated with a given ollision energy Etr and a given rovibrational state of CH2 hara terized by the quantum numbers J , K and v (v denotes globally the three numbers asso iated with the normal mode vibrations). By integrating the ross se tions σ(Etr , v, J, K) 9



over the Boltzmann distribution of the ollision energy Etr at a given temperature T , the rovibrational state-spe i rate oe ients k(T, v, J, K) an also be worked out. The omputational ampaign of traje tory al ulations onsisted of two blo ks. The rst blo k of al ulations was made of massive bat hes (2 millions traje tories for ea h temperature value) of QCT traje tories aimed at evaluating the fully thermalized QCT rate

oe ient k(T ) for a set of T values falling in the temperature interval ranging from 5 K to 1000 K. Forty four values of T were onsidered with a denser grid at low temperature. Thanks to the very large number of integrated traje tories, the statisti al error of the omputed rate

oe ient was always lower than 1%. It is worth noting that rotational states as high as

J = 23 non-negligibly ontribute (with a probability 1% larger than that of the most probable rotational state) at T = 1000 K. Moreover, rovibrational states of the rst ex ited vibrational level ( orresponding to the bending mode, whose vibrational frequen y is 1103 m−1 ) begins to ontribute starting from T = 220 K. The se ond blo k of al ulations was on erned with the omputation of the rovibrational state-spe i ross se tions by integrating 2 × 105 traje tories for ea h (Etr , v, J, K ) ase. We ran traje tories at 44 dierent ollision energies in the 0.001 − 9 k al mol−1 range, with a denser grid at low Etr . The number of (J, K ) rotational states was 70 for the ground vibrational state and 25 for the rst ex ited bending state.

3

RATE COEFFICIENTS

The al ulated values of the full thermal QCT rate oe ients k(T ) for the abstra tion H + CH2 → CH + H2 rea tion are shown in Figure 2. As apparent from the Figure and in

ontrast with the onstant values reported in the KIDA, UDfA and OSU databases, there is a lear in rease of the rate oe ient with temperature. As a matter of fa t, the k(T ) values omputed by us in rease signi antly with temperature up to T = 30 K. On the

ontrary, at higher temperatures their in rease with T be omes milder. In parti ular, from 10


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a value of 2.11 × 10−10 m3 s−1 at 10 K (similar to that of the KIDA and UDfA kineti s databases) they in rease up to that of the OSU database (at T = 260 K) and ontinue to in rease with a similar trend at higher temperature. The in reasing temperature dependen e of our QCT rates is less pronoun ed than that of the experiments of Fulle and Hippler (in luding their extrapolation to lower temperatures published by Hébrard et

al.) leading to

an underestimation of a fa tor 2 at T = 50 K and to an overestimation at T > 700 K. Besides the values of the QCT rate oe ient for the abstra tion H + CH2 → CH + H2 rea tion, Figure 2 shows also those for the ex hange H + CH2 → CH2 + H one. As apparent from the Figure, the latter rates display a dependen e on temperature qualitatively parallel to that of the abstra tion pro ess: a pronoun ed in rease at low temperature followed at

T > 30 K by a milder in rease. However, with respe t to the abstra tion urve, the value of the ex hange rate oe ient is smaller by a fa tor of about 2 − 4 and the in rease at high temperature is noti eably larger. This behaviour an be understood in terms of the

omplex-forming mi ros opi me hanism of the rea tion. As shown for the reverse CH + H2 → H + CH2 rea tion, 19,20 in fa t, when rea tants get loser they experien e the strong intera tion asso iated with the CH3 potential well during some pi ose onds. As a result, one (in the ase of the ex hange pro ess) or two (in the ase of the abstra tion pro ess) of the three C-H bonds an break and form the orresponding produ ts. To judge to whi h extent the system shows a omplex-forming me hanism, we estimated both abstra tion and ex hange rate oe ients by al ulating the probability of rea hing the potential well from rea tants and then applying a phase spa e theory 64 (PST) model for produ t formation. In the PST model, the statisti al probability for the omplex to rea t is obtained by weighting all available (energeti ally open) hannels equally, that is, as the ratio between the number of available rovibrational produ t states and the number of available rovibrational produ t and rea tant states, at xed total energy and total angular momentum quantum number. In parti ular, for ea h omplex-forming traje tory, the total energy and the total angular momentum are determined and, on e the latter is rounded to its nearest integer, its statisti al 11



probability is al ulated. The riterion used to dene the region of the strong intera tion inside the well is an energeti one 40 (more appropriate than the simpler empiri al radius of

apture used in ref 65). A

ordingly, traje tories experien ing along their path potential energy values more negative than −13.8 k al mol−1 (with zero energy orresponding to the asymptoti of the CH + H2 hannel) were onsidered omplex-forming traje tories. As a matter of fa t, up to about 200 K all abstra tion and ex hange pro esses pro eed, indeed, via the formation of the CH3 omplex. The estimated PST rate oe ients well reprodu e the QCT values, onrming the statisti al nature of the rea tion in the overall range of temperature investigated. The used model helps us to explain the dierent slope of the abstra tion and ex hange rates with the temperature. When the ollision energy is low, the number of rovibrational states of the H + CH2 hannel is small be ause of its endoergi ity (4.1 k al mol−1) whereas as the ollision energy in reases this number in reases faster than that of the CH + H2 hannel. This makes the dieren e between the lower temperature rates of abstra tion and ex hange pro esees larger than those at higher temperature ones. To gain more insight into the omplex-forming mi ros opi me hanism we analysed the

omplex lifetimes as a fun tion of the temperature. At 10 K, the omplex lifetime urve shows a maximum at very low value time (0.13 ps) and then it de reases quite slowly up to 5-6 ps. As the temperature in reases, the maximum of the omplex lifetime urve shift slightly to lower vales while it de reases faster. In brief, the formed omplex do not need mu h time to break statisti ally in both abstra tion and ex hange hannels. Furthermore, the analysis of the omplex lifetime of the inelasti pro esses onrms again the statisti al nature of the rea tion. In fa t, a long-lived omplex an break so that the hydrogen atom produ ed is the same that the initial one and therefore this event is lassied as inelasti . Most of the inelasti pro esses pro eed dire tly without omplex formation and through a

omplex of short lifetime. However, our results show that for traje tories that are trapped for more than 0.3 ps the three equivalent (in luding the inelasti one) CH + H2 hannels have the same probability. The same ee t was also observed for other barrierless rea tions. 12


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For example, for the O(1 D) + HF rea tion 66 the forward-ba kward symmetry hara teristi of statisti al rea tions is only observed for traje tories that are trapped for at least 100 fs. As mentioned above, the abstra tion rea tion is a pro ess exoergi by 4.1 k al mol−1 and the analysis of the partitioning of the available energy (in luding the exoergi ity) an be used to better understand the detailed me hanisms underlying the kineti s of the rea tion. This is, indeed, the ase of the dependen e of the average relative translational energy of the produ t diatoms and of their average rovibrational energies on the temperature shown in Figure 3. As apparent from the Figure, the average translational energy in reases near linearly with T from 4.2 k al mol−1 (at low temperature) to 6.4 k al mol−1 (at high temperature). However, by taking into a

ount the larger amount of energy available as the temperature in reases, the ratio of translational energy of the produ ts remains almost onstant (about 28%-30%). The dependen e of the energy released as vibration of both diatoms is also linear though atter with values varying from 3.5 to 4.3 k al mol−1 in the ase of CH and from 3.4 to 4.1 k al mol−1 in the ase of H2 . There is, also, a linear dependen e of the energy released as rotation of H2 (from 2.7 to 3.4 k al mol−1) on T . The ratio of the available energy going into these degrees of freedom de reases as T de reases from 23% to 20% for the vibration and from 18% to 16% for the rotation of H2 . On the ontrary, the energy released as rotation of CH in reases signi antly, from 1.0 to 2.7 k al mol−1 (i.e., from 6% to 13% of the total energy available). More information about the kineti s of the abstra tion rea tion omes from the detailed partitioning of energy into single degrees of freedom (rather than on the average). The distribution of the produ t translational energy is qualitatively similar at all values of temperature. It has, in fa t, its maximum between 1.8 and 4.5 k al mol−1 with a high probability at lower energy (at 0.1 k al mol−1 it an rea h more than 80% of the maximum for the lowest temperature while a the highest one it barely rea hes 30%) that de reases monotonously up to 20 k al mol−1. More quantitatively, when the distribution of the translational energy is tted using a Boltzmann expression, the temperature best reprodu ing the al ulated distri13



butions varies from 1240 K to 1840 K as the initial temperature in reases in the onsidered range. However, the tted Boltzmann distributions learly underestimate the al ulated high probability at low translational energy. As to the rovibrational energy of the diatomi produ ts, the most populated vibrational levels are the ground ones (almost ex lusively for H2 mole ule and with a very small fra tion in the rst vibrational state for CH). The distribution of the rotational states for the ground vibrational state of both mole ules at T = 10 an 500 K is shown in Figure 4. The al ulation of the rovibrational states using a semi lassi al method leads to ontinuous values and these values were boxed in intervals of 0.2 (the distributions obtained when using the gaussian binning pro edure were similar). The Figure shows that both rotational distributions of H2 peak near J ′ = 3 and extend to J ′ = 8 − 9 oering a rationale for the fa t that the average rotational energy shown in Figure 3 barely in reases with temperature. On the ontrary, the rotational distribution of CH learly shifts to high rotational states as the temperature in reases (the peak moves from J ′ = 3 to J ′ = 5 and populated states extend from J ′ = 12 to J ′ = 18 ) leading to a signi ant in rease of the released average rotational energy. To go into more detail, we analyse the ontribution to the thermal rate oe ient k(T ) of the rovibrational state-spe i rate oe ients k(T ; v, J). To this end, the state-spe i rate oe ients k(T ; v, J, K) were al ulated by integrating the orresponding ross se tions

σ(v, J, K) over the Boltzmann distribution at the required temperature. Next, they were weighted a

ording to the relative population of the K states for the rotational state J . Figure 5 shows the rovibrational state-spe i rate oe ients k(T ; v, J) and the thermal rate oe ients k(T ) as a fun tion of the temperature. As apparent from the Figure, the

k(T ; v, J) urves show a behaviour similar to that of the full thermal urve (a pronoun ed in rease at low temperature and a milder one above 30 − 40 K). Furthermore, the thermal

urve onse utively mat hes the omputed J spe i ones at in reasing values of J up to

J > 6 at the highest temperature. However, the most interesting feature of the plots is the non-monotoni trend of k(T ; v, J) with J and T at T < 220 K when the initial CH2 14


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radi al is in its ground vibrational state (v = 0). In fa t, the spe i rate for J = 0 is

learly larger than that for higher J values (only at temperatures lower than 10 K, values for J = 3 and 4 are larger though this is not appre iable in the Figure) and the urve for J = 4 falls above the one for J = 5 whi h in turns falls above the one for J = 6. On the ontrary, k(T ; v = 0, J = 1) is smaller than k(T ; v = 0, J ≤ 5) up to 100 K, and then it gradually in reases be oming larger than k(T ; v = 0, J > 1) above 200 K and analogously k(T ; v = 0, J = 2) is smaller than k(T ; v = 0, J = 4) below 200 K and than

k(T ; v = 0, J = 3) below 200 K. When the temperature is larger than 220 K, the higher ex itation the lower the state-spe i rate oe ient. Figure 5 shows also the state-spe i rate oe ients when the bending vibrational mode is ex ited (v = 1). It is worth pointing out here that the ontribution of these oe ients to the thermal rate oe ient begins to be not negligible at T > 220 K. As apparent from the Figure, the vibrational ex itation of CH2 leads to a lear de rease of the state-spe i rate oe ients. Furthermore, at T < 300 K the higher ex itation the larger the state-spe i rate oe ient while at higher temperatures the values tend to the opposite behaviour. A last omment is deserved to the ontribution of the K states to the state-spe i rate oe ients k(T ; v, J). For all the rovibrational states

onsidered, the value of the state-spe i rate oe ient de reases gradually as K in reases.

4

DETAILED PROPERTIES

The most detailed quantity al ulated in the present work is the ex itation fun tion for ea h (v, J, K) state. QCT estimates of the state-spe i ross se tions omputed for v = 0,

J = 0 − 6, K = 0 and v = 1 (i.e., the bending vibrational mode is ex ited), J = 0 − 3, K = 0 in the energy interval going from 0.001 to 9 k al mol−1 are plotted for omparison in Figure 6. As apparent from the Figure, the spe i feature of the plotted ross se tions is to be very large at low ollision energy and sharply de rease as energy in reases. When further in reasing the ollision energy the de reasing trend of the ross se tion ontinues though less 15



sharply. Again, the ex itation of the bending vibrational mode turns down the ross se tion to slightly smaller values and makes them onverge to the values al ulated at the vibrational ground state at high ollision energy. On the ontrary, the ee t of the rotational ex itation is small and not monotonous. In fa t, depending on the ollision energy, higher (or lower) values of rotational ex itation in rease (or de rease) the ross se tion. At the same time, the ee t of K for ea h rotational state (not shown in the Figure) is very small. The onstant de rease of the ross se tion with both ollision and internal energy seen in Figure 6 is typi al of barrierless rea tions. As a matter of fa t, in the absen e of barrier the rea tion is entirely driven by the potential energy gradient and any in rease of energy (either internal or translational) ountera ts the natural tenden y of the pro ess to be driven into the produ t hannel. This prompted further investigations of the dynami s of the ollision pro esses o

urring on the su h PES bearing a deep well. For this reason additional bat hes of traje tories were integrated at zero impa t parameter for an H atom approa hing along the axis Z to a non rotating CH2 mole ule sitting respe tively either on the XY (perpendi ular) or on the XZ and YZ ( oplanar) planes. The al ulations were performed at ollision energy of 0.02 k al mol−1 (a representative value of the low temperature astro hemistry pro esses). Su h low translational energy (together with the absen e of rotation and a null impa t parameter) fa ilitates the understanding of the nature of the steri intera tion. The results showed that perpendi ular (out of the XY plane) abstra tion ollisions are on the average about 15% more ee tive in rea ting than oplanar ones (the ones taking pla e on the XZ and YZ planes whi h have identi al probabilities). Of parti ular interest, however, is the detailed stru ture of the dependen e of the rea tive probability on the angle of atta k. To single out su h stru ture, the upper panels of Figure 7 show the (histogrammi ) plots of the rea tive probability for the abstra tion pro ess as a fun tion of the angle formed by the atta king (along the Z axis) H atom and the C-H bond of the mole ule. The rhs panel des ribes the

ase in whi h the CH2 mole ule lies on the XY plane (perpendi ular atta k) while the lhs 16


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panel des ribes the ase in whi h the CH2 mole ule lies on the YZ plane ( oplanar atta k). For both atta ks Z varies from the initial distan e (limited to 5 Å in the plot) and zero while the orientation angle of the CH bond varies from −180◦ to 180◦ . As apparent from the Figure, the probability of abstra tion is essentially the same (0.55) for all orientations of CH in perpendi ular atta ks. On the ontrary, in oplanar atta ks there is a lear modulation of the probability asso iated with the initial value of the CH orientation. A rationale for the dierent behaviour of perpendi ular and oplanar atta ks is provided in the lower panels of Figure 7 whi h show the ontour maps of the potential energy plotted as a fun tion of both the distan e of the atta king H atom and the orientation of CH2 for both the oplanar (lhs panel) and the perpendi ular (rhs panel) geometries when varying the distan e of H from CH2 while keeping the CH2 xed on its initial plane and optimizing both the bond lengths and the bond angle of CH2 . The Figure learly shows that the shape of the potential energy does not depend on the orientation of CH in the perpendi ular atta ks whereas it strongly depends on it when the atta k is oplanar. More detail on su h steri ee t on the dynami s of the ollision is given in the same panels of Figure 7 by plotting the initial part of two typi al abstra tion oplanar and perpendi ular traje tories: the latter do not re-orient CH while the former do it to the end of rea hing the well. The traje tories are plotted only up to the point in whi h they rea h the well. After this initial omplex-forming point, the traje tories wander inside the well assuming variable distorted CH3 four-atom geometries for several pi ose onds until produ t mole ules are formed, suggesting that the pro ess an be modelled using a statisti al treatment. 33,67,68

5

PARAMETRIZATION OF RATE COEFFICIENTS

Levering the on lusions of the previous Se tion, we are now able to a hieve the nal goal our work (i.e. to feed the already mentioned astro hemi al databases with a simple and su iently a

urate formulation of the dependen e of the abstra tion H + CH2 → CH 17



Page 18 of 37

+ H2 rate oe ient on the temperature) using an unied Arrhenius-like treatment for the al ulated rate oe ients shown in Figure 8 (temperatures higher than 70 K in lhs panel and lower than 70 K in rhs panel). As apparent from the Figure, rate oe ients at temperatures lower than 70 K and higher than 220 K show a lear Arrhenius behaviour that

an be parametrized respe tively as:

k(T ) = (2.50 × 10−10 cm3 s−1 ) exp(−1.67/T )

k(T ) = (3.02 × 10−10 cm3 s−1 ) exp(−30.36/T )

T = 5 − 70 K

(5)

T = 220 − 1000 K

(6)

From the unied Arrhenius-like approa h levering our dynami al al ulations we propose here the single overall double exponential formulation:

k(T ) = (2.50 × 10−10 cm3 s−1 ) exp(−1.67/T ) + (5.98 × 10−11 cm3 s−1 ) exp(−280.5/T ) (7) for temperatures ranging from 5 K to 1000 K. As shown in Figure 8, the proposed expression ts well the al ulated rate oe ients. For omparison we implemented also other formulations. In parti ular, we used the popular 1893 Kooij formula 69 (also known as the modied Arrhenius equation 70 ) and the most re ent one suggested by Aquilanti and Mundim 71,72 ( alled deformed Arrhenius), whi h bears a lear physi al meaning. 73 For the Arrhenius-Kooij parametrization:

k(T ) = α

T 300

β

h γi exp − T

(8)

the estimated best-tting parameters are α = 2.73 × 10−10 m3 s−1 , β = 0.060 and γ =

−0.678 K whereas for the Aquilanti-Mundim parametrization:

k(T ) = A

1−d

18

ǫ d1 T


(9)

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the estimated best-tting parameters are A = 2.95 × 10−10 m3 s−1 , ǫ = 33.4 K and d =

−10.2. Yet, as shown in Figure 8, however, both expressions do not seem to t a

urately our al ulated rate oe ients in the overall interval of temperature. In order to onform even with the simple formulation of for the astro hemi al databases, we an also divide the overall 5 − 1000 K interval of temperature in three subintervals: the rst one for temperature interval ranging from 5 to 70 K (i.e. temperature values relevant to the outer envelop of the star forming regime), the se ond one for the temperature interval ranging from 70 to 220 K, (i.e. temperature values relevant to the hot ore region of protostars) and the third one for the temperature interval ranging from 220 to 1000 K, (i.e. temperature values relevant to des ribe the ir umstellar envelops). The al ulated rate oe ients in the rst and third range of temperature were parametrized using the mentioned simple Arrhenius formula (see eqs 5 and 6) whereas the intermediate range of temperature were parametrized using the Arrhenius-Kooij formula with α = 2.69 × 10−10

m3 s−1 , β = 0.097 and γ = −3.27 K. As apparent from the left panel of Figure 8, this parametrization leads to an optimal t to the al ulated values.

6

CONCLUSIONS

Using the Open S ein e Cloud s heme of MOSEX we formulated for some popular databases (i.e. using simple analyti al expressions) the temperature dependen e of the a

urately determined values of the rate oe ients trying to provide a theoreti al ba kground to some available experimental data from 1000 K down to T = 5 K and surmounting the fa t that not only experimental data are s ar e but they are also often assumed to be temperature independent for a wide range of T values. Under these ir umstan es we performed quasi lassi al dynami s omputations by adopting omputational s hemes keeping approximations and statisti al treatments under severe ontrol. This has been applied to the determination of the quasi lassi al formulation of the initial states of the rea tants, to the adoption of 19



the number of integrated traje tories, to the numeri al te hnique used for evaluating the potential energy derivatives, to the size of the time integration step, et . The rst lear eviden e obtained from dynami al al ulations performed at the above mentioned level of a

ura y was the strong dependen e of the rate oe ient on the temperature, mainly a low temperature, in ontrast with the onstant value adopted in most of the kineti s astro hemi al databases. The se ond lear eviden e is the dramati ally dierent dynami al behaviour of the abstra tion and ex hange rea tive me hanisms. This auses a dieren e of a few hundred per ents in the related whole rea tive rate oe ients over the investigated temperature interval and a dieren e in the related energy ee tiveness in promoting rea tion and determining the energy partitioning among produ t states. Another lear eviden e is the arising of a denite steri ee t (with respe t to the ollision of the in oming H atom and the angle formed with the CH bond of the target mole ule) in

oplanar ollisions (before entering the near trapped regime asso iated with the deep short distan e well) as opposite to the isotropi behaviour of perpendi ular ollisions. The possibility of segmenting the dynami ee t of the rea tive atta k in a longer range

omponent (dominated by the dynami s indu ed by the attra tive tail of the potential) and a shorter range one (dominated by the statisti s regime asso iated with the near trapping at the well) allows in any ase to more learly work out a single double-exponential Arrheniuslike formulation of the rate oe ient for the whole 5 − 1000 K temperature interval.

A knowledgement E.G. and P.G.J. a knowledge nan ial support from the MINECO/FEDER of Spain under grants CTQ2015-65033-P and PGC2018-096444-B-I00. P.G.J. a knowledges funding by Funda ion Salaman a City of Culture and Knowledge (programme for attra ting s ienti talent to Salaman a). Thanks are also due to the Oklahoma University Super omputing Center for Edu ation & Resear h (OSCER) and the European Grid Infrastru ture (EGI) through 20


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COMPCHEM Virtual Organization for providing omputing resour es and servi es.

21



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2019, 7, 380.

30


Front. Chem.

1.0

T=10 K

0.8

Probability

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


1.0

T=50 K

0.8

0.6

0.6 Classical Quantum

0.4

Classical Quantum

0.2 0.0

Probability

Page 31 of 37

0.4 0.2

0

1

2

J

3

4

0

5

1

2

J

3

4

5

0.0

Figure 1: Classi al and quantum distribution of rotational states at T =10 K and 50 K.

31



3.0 2.5

k /1010 cm3 s-1

KIDA, UDfA OSU Fulle, Hippler Hebrard et al QCT Abstraction QCT Exchange

2.0 1.5 1.0 0.5 0

100

200

300

400

500

T /K

600

700

800

900

1000

Figure 2: QCT abstra tion and ex hange full thermal rate oe ients for the H + CH2 rea tion plotted as a fun tion of the temperature. Statisti al errors are smaller than the size of the used symbols. Values quoted in kineti s databases and experimental results are also shown for omparison.

Graphi al TOC Entry H + CH2

CH + H2

3.0

k /1010 cm3 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

Page 32 of 37

2.5

k(10 K)= 2.11 cm3 s-1

2.0 KIDA, UDfA OSU Fulle, Hippler Hebrard et al QCT Abstraction

1.5

1.00

100

200

300

400

500

600

T /K

700

800

900 1000

32


Page 33 of 37

6

Relased energy /kcal mol-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


5

Etr Evib(CH) Erot(CH) Evib(H2) Erot(H2)

4 3

2

10

100

200

300

400

500

T /K

600

700

800

900

1000

Figure 3: Translational average energy released in the H + CH2 → CH + H2 rea tion and vibrational and rotational average energy of the CH and H2 produ ts plotted as a fun tion of the temperature.

33



100

J’(CH) T=10 K J’(CH) T=500 K J’(H2) T=10 K J’(H2) T=500 K

80

% Probability

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

Page 34 of 37

60

40

20

0

0

2

4

6

8

10

12

Rotational state, J’

14

16

18

Figure 4: Distribution of rotational states of the ground vibrational states of CH and H2 diatoms formed in the H + CH2 → CH + H2 rea tion at 10 K and 500 K.

34


Page 35 of 37

3.5

3.0

k /1010 cm3 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


v=0,J=0 v=0,J=1 v=0,J=2 v=0,J=3 v=0,J=4 v=0,J=5 v=0,J=6 v=1,J=0 v=1,J=1 v=1,J=2 v=1,J=3 Thermal

2.5

2.0

1.5 0

100

200

300

400

500

T /K

600

700

800

900

1000

Figure 5: Rovibrational state-spe i rate oe ients for the H + CH2 → CH + H2 rea tion plotted as a fun tion of the temperature. Thermal rate oe ients are also shown for

omparison.

35



v=0,J=0 v=0,J=1 v=0,J=2 v=0,J=3 v=0,J=4 v=0,J=5 v=0,J=6 v=1,J=0 v=1,J=1 v=1,J=2 v=1,J=3

200

150

250

200

150

100

Cross section /Å²

250

Cross section /Å²

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

Page 36 of 37

50 100 0

0 1 2 3 4 5 6 7 8 9 0.00 -1

Etr /kcal mol

0.01 0.02 0.03 0.04 0.05

Etr /kcal mol-1

Figure 6: Rovibrational state-spe i ross se tions for the H + CH2 → CH + H2 rea tion plotted as a fun tion of the temperature. Right panel shows a zoom at low energy.

36


0.7 0.6

Probability

0.7 0.6 0.5 0.4 -135

-90

-45

0

90

45

135

0.5 0.4 0.3 -180

180

-135

-90

-45

0

45

90

135

180

5

5

4

0

4

0

3

−5

3

−5

2

−10

2

−10

1

−15

1

−15

0 −180

−135

−90

−45

0

45

90

135

180

Z/Å

0 −180

−20

−135

−90

−45

Orientation angle

0

45

90

135

V / kcal/mol

5

5

V / kcal/mol

Z/Å

0.3 -180

−20

180

Orientation angle

Figure 7: Upper panels: Probability of the abstra tion rea tion as a fun tion of the angle formed by CH and the initial relative velo ity when CH2 is either oplanar (lhs panel) or perpendi ular (rhs panel) to the atta king H atom. Lower panels: Contour maps of the potential energy plotted as a fun tion of the distan e between the atta king H atom and the C atom and of the the orientation angle for both CH2 oplanar and perpendi ular. The zero energy is pla ed in the asymptoti CH + H2 hannel and isoenergeti ontours are plotted every 2.5 k al mol−1. On the ontour maps a typi al traje tory for both oplanar and perpendi ular atta k is drawn.

-22.1

-21.92

T < 70 K

QCT Double exponential Arrhenius-Kooij Aquilanti-Mundim Arrhenius-Kooij 70-220

-22.00 -22.04

QCT Double exponential Arrhenius-Kooij Aquilanti-Mundim

-22.08

-22.3

-22.4

-22.12 -22.16 0.000

-22.2

ln (k / cm3 s-1)

T > 70 K

-21.96

ln (k / cm3 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


Probability

Page 37 of 37

0.002

0.004

0.006

0.008

T-1 /K-1

0.010

0.012

0.014

0.02

0.04

0.06

0.08

0.10

0.12

T-1 /K-1

0.14

0.16

0.18

Figure 8: Parametrization of the al ulated QCT rate oe ients.

37


0.20

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