Theoretical Study of the Reactions of Methane and Ethane with

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Article

Theoretical Study of the Reactions of Methane and Ethane With Electronically Excited N(A# ) 2

3

u+

Alexander S. Sharipov, Boris I. Loukhovitski, and Alexander M. Starik J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04244 • Publication Date (Web): 06 Jun 2016 Downloaded from http://pubs.acs.org on June 9, 2016

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The Journal of Physical Chemistry

Theoretical Study of the Reactions of Methane and Ethane with Electronically Excited N2(A3Σu+) Alexander S. Sharipov, Boris I. Loukhovitski and Alexander M. Starik* Central Institute of Aviation Motors, Moscow, Russia Scientific Educational Centre “Physical-Chemical Kinetics and Combustion”, Moscow, Russia

*Corresponding author. +7 495 3616468. E-mail: [email protected] Abstract Comprehensive quantum chemical analysis with the usage of density functional theory and postHartree-Fock approaches were carried out to study the processes in the N2(A3Σu+) + CH4 and N2(A3Σu+) + C2H6 systems. The energetically favorable reaction pathways have been revealed on the basis of the examination of potential energy surfaces. It has been shown that the reactions N2(A3Σu+) + CH4 and N2(A3Σu+) + C2H6 occur with very small or even zero activation barriers and, primarily, lead to the formation of N2H + CH3 and N2H + C2H5 products, respectively. Further, the interaction of these species can give rise the ground state N2(X 1Σg+) and CH4 (or C2H6) products, i.e. quenching of N2(A3Σu+) by CH4 and C2H6 molecules is the complex two-step process. The possibility of dissociative quenching in the course of the interaction of N2(A3Σu+) with CH4 and C2H6 molecules has been analyzed on the basis of RRKM theory. It has been revealed that, for the reaction of N2(A3Σu+) with CH4, the dissociative quenching channel could occur with rather high probability, whereas in the N2(A3Σu+) + C2H6 reacting system analogous process was little probable. Appropriate rate constants for revealed reaction channels have been estimated by using a canonical variational theory and capture approximation. The estimations showed that the rate constant of N2(A3Σu+) + C2H6 reaction path is considerably greater than that for the N2(A3Σu+) + CH4 one.

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1. Introduction For past years, a great deal of interest has been expressed to the analysis of the role of electronically excited species in different reacting systems.1-6 Molecules, excited to metastable electronic states, are potentially of great importance for gas-phase chemistry, because they can possess higher reactivity compared with non-excited ones. Nitrogen molecule in the lowest triplet electronic state, N2(A3Σu+), has rather high radiative lifetime (~ 1 s), because the radiative transition to the ground state N2(X 1Σg+) is spin forbidden.7,8 At the same time, the electronic excitation energy of N2(A3Σu+) molecule (Te = 6.22 eV)8,9 is greater than the typical bond energy for the most di- and polyatomic molecules, including hydrocarbon ones. Elementary processes, involving metastable N2(A3Σu+) molecule, come into play in the upper and middle atmosphere of Earth and of Saturnian satellite Titan,10-15 in the hypersonic flows under re-entry conditions,16,17 in the nitrogen and nitrogen-oxygen discharge plasma18-22 and in plasma chemistry.23,24 It should be emphasized that the phosphorescence at the transition A3Σu+→ X 1Σg+ of N2, corresponding to the Vegard-Kaplan band system, was well studied in the past both experimentally and theoretically.9,25 The emission of Vegard-Kaplan band is important for aurora and laboratory investigations of the processes with active nitrogen. It should be emphasized that the reaction kinetics of N2(A3Σu+) molecule have not been well established up to now, despite the great interest expressed by the researchers engaged in the field. So, the processes of electronic-electronic and electronic-vibrational exchanges, involving excited N2(A3Σu+) molecule, as well as its quenching were extensively investigated both experimentally7,8,22,26-32 and theoretically.15,31-33 However, in the most cases, only the decay of N2(A3Σu+) molecule was detected, and detailed data on the rate constants for the chemical exchange reactions with N2(A3Σu+) molecule, are very scarce. Meanwhile, anomalously high chemical reactivity of N2(A3Σu+) was discussed as far back as 1970s.7 Special interest was expressed recently to the reactions of N2(A3Σu+) molecule with saturated hydrocarbons such as CH4 and C2H6. Attention to these processes is caused by their possible crucial role 2 ACS Paragon Plus Environment

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for plasma-assisted combustion23,24 and for plasma-chemical fuel reforming.20,34-37 It has been assumed that the reactions of N2(A3Σu+) with CH4 and C2H6 lead to collisional quenching of N2(A3Σu+)8,26,27 or to the dissociation of hydrocarbon molecules with simultaneous quenching of N2(A3Σu+) (dissociative quenching).13,20,21,23,36 However, these suppositions were done without sufficient proofs based on the investigation of the potential energy surfaces for the N2(A3Σu+) + CH4 and N2(A3Σu+) + C2H6 systems. The present study is aimed at theoretical analysis of possible reaction paths in such systems on the basis of quantum chemical calculations and estimations of rate coefficients of corresponding channels.

2. Computational Details 2.1 Investigation of Potential Energy Surfaces Analysis was based on series of quantum chemical computations with the use of density functional theory (DFT). It is worth noting that, though multireference methods are usually used for the theoretical exploration of reactions with electronically excited molecules,5 DFT approach can also be successfully applied to study such type of reactions.4 So, the Becke’s hybrid half-and-half density functional BH&HLYP,38 proved to be very efficient computational tool for the estimates of energy barrier for exchange reactions,39-41 was chosen to explore the lowest singlet and triplet potential energy surfaces (PESs) of N2 + CH4 and N2 + C2H6 systems. The computations showed that the BH&HLYP functional predicted proper value of the electronic excitation energy of N2(A3Σu+) molecule (Te = 6.22 eV)8, in contrast to the other popular DFT functionals. So, the BH&HLYP functional with the aug-cc-pvQZ basis set provides Te = 6.31 eV, whereas the other widely applied functionals, B3LYP, O3LYP, X3LYP, B97-2, BMK and M06, give Te value in the range of 6.5-6.7 eV. In our computations, the geometry of reactants, transition states and possible products of reaction pathways were optimized at the UBH&HLYP/aug-cc-pvDZ level of theory. The characterization of each stationary point either as a minimum or as a first-order saddle point was conducted by using a vibrational frequency analysis at the same level of theory. The computed vibrational frequencies were multiplied by an appropriate scaling factor equal to 0.93 in line with the 3 ACS Paragon Plus Environment


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recommendations of Merrick et al.42 This allows one to compensate an incomplete treatment of electron correlation as well as the effect of truncated basis set and the neglect of the anharmonicity of molecule vibrations. The minimum energy paths (MEPs) for the reactions under study were obtained by using the Gonzalez-Schlegel method.43 To define the energy values of critical points on PESs with the use of BH&HLYP functional more accurately, the three-point basis set extrapolation of the electronic energy to the complete basis set (denoted herein and hereafter as ‘aug-cc-pv∞Z’) was applied in the same manner as in our previous study.44 In doing so, the single point calculations with the Dunning’s aug-cc-pvTZ and aug-cc-pvQZ basis sets45 were performed for all critical points. The electronic energy values, calculated at the UBH&HLYP/aug-cc-pv∞Z level of theory, were supplemented with zero point energy (ZPE) correction. Additionally, in order to refine the energy of activation barriers, the CCSD(T)46 and QCISD(T)47 methods as well as the composite ones, G2MP2,48 G3MP249 and G4MP2,50 developed especially to achieve high target accuracy, were applied. All quantum chemical calculations were conducted by using Firefly QC program package51 which is partially based on the GAMESS(US) source code.52 Visualization of computational results was performed by using ChemCraft software.53

2.2 Calculation of Reaction Rate Constants In the case when there is a pronounced saddle point on the MEP of the reaction, the rate constant was estimated with the usage of canonical variational theory (CVT) with Eckart-type tunneling correction in the same manner as in our previous works.44,54 The projected vibrational frequency analysis along the reaction path,55 needed for the CVT estimations, was conducted at the UBH&HLYP/aug-cc-pvDZ level of theory. Note that precisely this level of theory was utilized during the MEPs calculations. When calculating the vibrational partition functions for the reactants and transition states, the “effective” harmonic oscillator model56 was applied. In doing so, it was assumed that the partition 4 ACS Paragon Plus Environment

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function for vibrational modes with the frequencies lower than the certain cutoff value (ωc = 100 cm−1) was equal to the partition function of harmonic oscillator calculated for the frequency ωc. This allowed us to avoid possible overestimation in the values of vibrational partition function for the modes with low frequencies due to numerical errors arising upon typical harmonic frequency analysis based on the differentiation of the PES in the points of energy minima.56,57 For the internal degrees of freedom of a molecule, that are specified as internal rotation, the approximate model of Truhlar, providing a smooth approximation for the partition function from free rotator to harmonic oscillator,58 was used. The heights of corresponding torsional barriers were calculated by using the relaxed potential scans in the vicinity of critical points. In order to estimate the rate constant for the barrierless reactions, for which the MEP does not comprise saddle point, the capture approximation in the manner, reported in the works,44,54 was applied. In doing so, the interaction of two polar particles at the distance r was described by the effective spherically symmetrical potential

 C 6ind C 6el C eff + ϕ eff (r ) = − 66 , C 6eff = C 6disp 1 + disp r 4C 6disp  C6

2

  , 

(1)

involving the dispersion C6disp, electrostatic C6el and polarization C6ind interaction terms. The electric properties, required for the calculations of these terms, such as dipole moment and static polarizability, were estimated by using the UBH&HLYP/aug-cc-pvTZ level of theory. The temperature-dependent rate constant k(T) of bimolecular barrierless reaction between two particles A and B for the potential (1) can be expressed as

(

)

1

 C 6eff 2 k bT  6 p   , k (T ) ≅ 1.706  p p  M3  PES e A B e e

(2)

where kb is the Boltzmann constant, T is the gas temperature, M is the reduced molecular mass of colliding molecules, peA, peB and pePES are the electronic degeneracies of reactants (A and B) and reactive PES, respectively.


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3. Results and Discussion 3.1 Potential Energy Surfaces and Reaction Pathways The results of the calculations of reaction pathways, which connect the reactants and products of the processes N2(X 1Σg+) + CH4 and N2(A3Σu+) + CH4, as well as N2(X 1Σg+) + C2H6 and N2(A3Σu+) + C2H6, are schematically depicted in Figure 1. Herein and hereafter, the following designations for critical points on PESs are used: SP for saddle points, IM for local minima, and the left superscripts denote the multiplicity. Structures, frequencies of normal vibrations and rotational constants for saddle points and minima are given as Supporting Information. One can see that the reaction of N2(A3Σu+) with CH4 leads to the formation of doublet N2H(X 2A′) and CH3(X 2A2′′) products through the saddle point 3SP which lies only slightly lower (by ~0.03 eV) than the reactants on the triplet PES. Thus, the UBH&HLYP/aug-cc-pv∞Z level of theory predicts that this reaction is a barrierless one. In addition, BH&HLYP calculations revealed the existence of weakly bound N2(A3Σu+)···CH4 complex (3IM) on the reactant side from the saddle point. 3 +

N2(A Σu)+CH4 -1 -2

(0.01) 3 +

3 3

0.02

(a)

(0.02)

SP

IM

0

3

(0.03)

-2

N2H + CH3

-3 -4

1

-5 -6 -7

(0.01)

CH3N2H( A) 1 +

(0.01)

N2(X Σg )+CH4

N2(A Σu)+C2H6

(0.01)

(b)

-1

(0.03)

CH3N2H( A)

E, eV

0

E, eV

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3

(0.01)

1

(0.04)

C2H5N2H( A)

(0.03)

N2H + C2H5

-3

C2H5N2H( A)

-4 -5 1 +

-6

(0.01)

N2(X Σg )+C2H6

-7

Figure 1. Triplet and singlet (solid and dashed lines) electronic terms for the N2 + CH4 (a) and N2 + C2H6 (b) systems obtained at the UBH&HLYP/aug-cc-pv∞Z level of theory with ZPE correction. The values of CCSD T1 diagnostics are given in superscript brackets.

As to the reaction of N2(A3Σu+) with C2H6, it leads to the barrierless formation of N2H(X 2A′) and C2H5(X 2A′) radicals. However, the corresponding MEP, obtained at the applied level of theory, is


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monotonous without local minimum. Thus, the following reaction pathways with excited N2(A3Σu+) molecule were revealed in the systems under study: N2(A3Σu+) + CH4 → N2H + CH3,

(R1)

N2(A3Σu+) + C2H6 → N2H + C2H5.

(R2)

The corresponding MEP profiles on the triplet PESs as well as the energy profiles, resulting from the projection of the triplet MEPs on the singlet PESs for the reactions (R1) and (R2), are shown in Figure 2. One can see that the triplet and singlet PESs are degenerated at the product regions, i. e. the interaction of excited N2(A3Σu+) molecule with CH4 and C2H6 leads to the formation of the same products as in the case of the reaction of ground state N2(X 1Σg+) molecule with methane and ethane. This fact suggests that the products of reaction paths (R1) and (R2), in turn, can react with the production of the ground state N2(X 1Σg+) molecule via the following channels N2H + CH3 → N2(X 1Σg+) + CH4 ,

(R3)

N2H + C2H5 → N2(X 1Σg+) + C2H6. (R4) Note that, at the applied level of theory, these processes proved to be barrierless ones. 1

1

(a)

3 +

N2(A Σu) + CH4

0

0

-1

-1

E, eV

E, eV

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N2H + CH3

-2 -3 -4 -5 -2

0

2

4 0.5

qr, amu Å

-2

N2H + C2H5

-3 -4

1 +

N2(X Σg)+CH4 6

8

(b)

3 +

N2(A Σu) + C2H6

1 +

N2(X Σg ) + C2H6

-5 0

2

4

6

8

0.5

qr, amu. Å

Figure 2. Electronic energy profiles as a function of the reaction coordinate qr along the MEPs for the N2(A3Σu+) + CH4 (a) and N2(A3Σu+) + C2H6 (b) systems obtained at the UBH&HLYP/aug-cc-pvDZ level of theory (solid curves) as well as the energy profiles resulting from the projection of triplet reaction path on the singlet PESs (dashed curves). Pointwise electronic energy values, predicted at the UCCSD(T)/aug-cc-pvDZ level of theory, are depicted by symbols. ACS Paragon Plus Environment

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However, there exists a possibility that the N2H and CH3 as well as N2H and C2H5 reactants can also form the stable molecules, such as methyl diimide (CH3N2H) and ethyl diimide (C2H5N2H), respectively, via the pathways N2H + CH3 → CH3N2H,

(R5)

N2H + C2H5 → C2H5N2H.

(R6)

Due to degeneracy of triplet and singlet PESs, these molecules can arise both in the singlet and in the triplet electronic states (see Figure 1). Note that the structure and basic properties of CH3N2H molecule was established rather well both experimentally59-61 and theoretically,62 whereas there has been a little information for the C2H5N2H molecule. From the plots shown in Figure 1 it is seen that the usage of BH&HLYP functional at the basis set limit allowed one to reproduce with reasonable accuracy the reaction enthalpies ∆Hr0 of the processes under study. So, the computed values of ∆Hr0 for the reaction channels (R3) and (R4) are equal to -4.45 and -4.27 eV, whereas the database of Baulch et al.63 gives for these channels ∆Hr0 = 4.83 eV and ∆Hr0 = -4.66 eV, respectively. The BH&HLYP functional also predicts properly the value of electronic excitation energy of N2(A3Σu+) molecule, comparable with that calculated by highly accurate approaches64,65 (the obtained value of Te was equal to 6.24 eV, whereas the studies64,65 gave Te in the range of 6.13-6.22 eV). The energy of CH3N2H(1A) dissociation into CH3 and N2H products, obtained in the present work (De = 1.87 eV), is also close to the data of Vidyarthi et al.66 (De was in the range of 1.95-2.15 eV). Thus, it was revealed that the interaction of N2(A3Σu+) with CH4 or C2H6 molecules occur, primarily, through the H-abstraction reaction with small or even zero activation barrier and resulted in the formation of N2H molecule and methyl or ethyl radical, respectively. From Figure 2 it follows that the triplet and singlet PESs for the reactions (R1) and (R2) come close to each other, at least, along the triplet MEP, only in the vicinity of the degenerating PESs region. However, at the corresponding values of reaction coordinate (qr ≈ 2-3 amu0.5Å), the energy of the nuclei motion along this coordinate is rather


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large, and, in accordance with Landau-Zener model,67,68 the probability of non-adiabatic transition between triplet and singlet PESs, at such qr values, must be small. In general case, the triplet and singlet PESs can come close to each other or even intersect at the other regions of PES apart from the MEP. However, in this case, non-adiabatic process can occur with non-zero activation barrier. Therefore, one can conclude that the reactive channels (R1) and (R2) prevail over possible collisional quenching of N2(A3Σu+) by CH4 and C2H6 molecules due to intersection of corresponding PESs. However, the sequences of the (R1) and (R3) as well as (R2) and (R4) processes can be attributed to the two-step collisional quenching of N2(A3Σu+) by CH4 and C2H6 molecules. Note, that the complete exploration of possible intercrossing of these PESs is beyond the scope of the present work. As was mentioned above, the UBH&HLYP/aug-cc-pv∞Z level of theory predicts that the processes (R1) and (R2) are barrierless. However, it is known that DFT methods have a tendency to underestimate slightly the value of activation barrier for the exchange reactions.39,69 In order to verify the accuracy of calculated electronic energy profiles, the unrestricted coupled-cluster method including single and double substitutions with a quasi-perturbative triples contribution (UCCSD(T)) method, proved to be very accurate for the estimation of the energy barrier,70,71 was also applied. The energy values, obtained in the course of pointwise UCCSD(T)/aug-cc-pvDZ calculations along the corresponding MEPs, are depicted in Figure 2. It is seen that the electronic energy profiles, calculated at different levels of theory, are similar. However, CCSD(T) computations suggest that the elementary processes (R1) and (R2) occur with small, but non-zero activation barriers, which are equal to ~0.05 and ~0.01 eV, respectively. In addition, to verify whether the single determinant Hartree-Fock reference wave function describes the systems under study satisfactorily, the coupled-cluster calculations were examined by using the T1 diagnostics.72,73 As is known, the larger the T1 value is, the less reliable are the results of the single-reference coupled cluster wave function. The results of this diagnostics are depicted in figure 1. One can see that, in line with our calculations, the values of Euclidean norm of the T1 vector for the 9 ACS Paragon Plus Environment


reactants, saddle points and products for the processes (R1)-(R4) are within the range 0.01-0.03. As far as the traditional limit of the reliability for single reference coupled cluster calculations, proposed by Lee and Taylor,72,73 for T1 is equal to 0.02, the obtained range for the T1 values indicates that the systems under study are of moderate multireference character. However, for the open-shell systems, the reliability limit can be greater than T1 = 0.02. In fact, even the T1 values of 0.04–0.05 are frequently accepted as reliable one for reaction paths.74,75

1000 500 0

1500

(a)

UCCSD(T) UQCISD(T)

500

3 +

N2(A Σu )+CH4 3

-500

IM

3

SP

-1000

(b)

UCCSD(T) UQCISD(T)

1000

Ea, K

1500

Ea, K

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3 +

N2(A Σu)+C2H6

0 -500

-1000

-1500

-1500 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

0.5

qr, amu Å

qr, amu Å

Figure 3. The pointwise electronic energy values, calculated at the UCCSD(T)/aug-cc-pvDZ and UQCISD(T)/aug-cc-pvDZ levels of theory (symbols), along the initial parts of triplet MEPs (solid curves) for the N2(A3Σu+) + CH4 (a) and N2(A3Σu+) + C2H6 (b) systems.

In order to determine the values of activation energy Ea for the reactions (R1) and (R2) more precisely, the series of pointwise CCSD(T) and QCISD(T) calculations along the initial parts of triplet MEPs for these processes, obtained at the UBH&HLYP/aug-cc-pvDZ level of theory, were performed. The results of these calculations are shown in Figure 3. Activation energies Ea, determined by using three-point parabolic interpolation from single point computations as well as those, corrected for ZPE, are presented in Table 1. It is worth noting that the applied procedure is a version of well-known IRCMax approach,76 in which one selects the maximum energy points obtained at the high level of theory along the low-level calculations of MEP. The magnitudes of Ea, obtained by using composite computational schemes, G2MP2, G3MP2 and G4MP2, are also given there. One can conclude that the ACS Paragon Plus Environment

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refined value of Ea for the reaction of N2(A3Σu+) with CH4 are within the range of 700-1000 K, whereas the activation energy for the reaction of N2(A3Σu+) with C2H6 does not exceed 120 K or may be even negative. Though, the systematic computations with accurate multireference electron correlation procedure are beyond the scope of the present work, some additional calculations for the (R1) and (R2) reactions paths with the use of the multireference theory were carried out. The series of energy calculations along the initial parts of the corresponding MEPs were done with the use of extended multi-configuration quasi-degenerate second order perturbation theory (XMCQDPT2) based on the CASSCF reference wave function.77 In doing so, the following active space configurations were taken into consideration: 12 electrons and 9 molecular orbitals for the process (R1) and 16 electrons and 11 molecular orbitals for the process (R2). The resulting values of Ea, estimated with the use of aug-cc-pvTZ basis set, are presented in Table 1. One can see that the Ea values, obtained with the use of the multireference secondorder Møller−Plesset (MRMP2) theory, are substantially higher than those predicted by coupled cluster methods. However, the CCSD(T) method is rather preferred for the calculations of the activation barriers of reactions than MRMP2 one for the problems of moderate multireference character.78,79 Moreover, the CCSD(T) estimates of energy values coincide rather well with the data obtained by composite methods. Therefore, the Ea values predicted by XMCQDPT2 method, can not be accepted as reliable for the reactions under study. It is worth noting that the results of our quantum chemical calculations, indicating that the activation barrier for the R2 process is lower than that for the (R1) one, correlates with the expectations based on some known empirical rules. So, in accordance with the Evans-Polanyi-Semenov relationship,80,81 for a number of homologous atom abstraction reactions, the activation energy is dependent linearly on the reaction enthalpy ∆Hr0 : E a = E 0 + γ∆H r0 ,

(3)


11


where E0 and γ are the empirical parameters. Our calculations gave Hr0(R1) = -1.782 eV, ∆Hr0(R2) = 1.969 eV (see figure 1,). Assuming that γ = 0.25 for exoergic reactions,81 one can obtain Ea(R1) Ea(R2) = 0.05 eV ( ≈ 600 K). This magnitude coincides rather well with the analogous value borrowed

from the data presented in Table 1. As well, the obtain data on Ea values for the channels (R1) and (R2) conform to the empirical correlation between the activation energy and polarizability of reactants, justified for the homologous atom abstraction reactions.82 So, as far as the magnitude of Ea is inversely proportional to the static polarizability of reactant, the activation barrier for the reaction of N2(A3Σu+) with CH4 (α = 2.43 Å3) must be higher than that for the reaction of N2(A3Σu+) with C2H6 (α = 4.16 Å3).82 The dependence of molecule reactivity on the α value is based on the fact that the attractive dispersion forces, exerting between reactants and competing with the repulsive covalent forces, are dependent on the reactant polarizability. 1000 0

UCCSD(T) UQCISD(T)

N2H + CH3

1000

(a)

UCCSD(T) UQCISD(T)

N2H + C2H5

0 -1000

-2000

-2000

(b)

Ea, K

-1000

Ea, K

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-3000

-3000

-4000

-4000

-5000

-5000

-6000

-6000 0

2

4

6

8

10

0

2

0.5

4

6

8

10

0.5

qr, amu Å

qr, amu Å

Figure 4. Electronic energy profiles as a function of the reaction coordinate qr along the initial parts of singlet MEPs for the N2H + CH3 (a) and N2H + C2H5 (b) systems defined at the UBH&HLYP/aug-ccpvDZ level of theory (curves) as well as the pointwise electronic energy values obtained at the UCCSD(T)/aug-cc-pvDZ and UQCISD(T)/aug-cc-pvDZ levels of theory (symbols).

In order to investigate more thoroughly the MEPs for the barrierless processes (R3) and (R4), the initial parts of MEP profiles on the singlet PESs, obtained with the UBH&HLYP/aug-cc-pvDZ level of


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theory, were refined during the series of pointwise CCSD(T) and QCISD(T) calculations along these paths. The results of these calculations are depicted in Figure 4. As is seen, both DFT and high-level post-Hartree-Fock calculations exhibited that the MEPs for the processes (R3) and (R4) do not comprise saddle points.

3.2 Reaction Kinetics Quantum chemical calculations showed that the reaction of N2(A3Σu+) with CH4 occurred with small activation barrier and led to the formation of N2H molecule and methyl radical. The rate constant of this process was estimated by using CVT. In order to estimate the possible uncertainty in the value of this rate constant, associated with the inaccuracy in the quantum chemical calculations of activation barrier, different Ea values, listed in Table 1, were considered. The obtained temperature-dependent rate coefficients are depicted in Figure 5. It is seen that, at T = 300 K, the magnitude of the uncertainty for the rate coefficient of channel (R1) mounts to a factor of 3, whereas, at high temperatures (T = 25003000 K), it does not exceed 15%.

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calculation of energy barriers: CCSD(T) (dashed curves), QCISD(T) (dotted curves), composite methods G4MP2 for the path (R1) and G2(3)MP2 for the path (R2) (solid curves).

It would be helpful to compare the calculated rate constant with the data for the reaction of N2(A3Σu+) with CH4, reported by other researchers. Note, that until now it has been believed that this reaction is associated either with collisional or with dissociative quenching of N2(A3Σu+). So, the experimental data on the collisional quenching of N2(A3Σu+) by CH4 were reported by Slanger et al.26,27 It was obtained that the value of the rate constant at T = 300 K is in the range of 2×109 – 4×109 cm3mol1 -1

s . However, only the rate of N2(A3Σu+) decay was detected in these experiments. As it was shown

above, the reactive channel (R1) must prevail over direct collisional quenching, and, therefore, the data reported by Slanger et al.26,27 can be attributed to the rate constant of the process (R1). The comparison of the predictions with measurements demonstrates that the calculated value of the rate constant for the process (R1), even with maximal energy barrier among those presented in Table 1 (Ea = 1015 K), is substantially greater at low temperatures (T < 400 K) than the data reported elsewhere26,27 (by a factor of 9-10). It should be noted that the application of more rigorous methods than CVT for the evaluation of rate constant, such as microcanonical variational transition state theory (µVT), implying summation of micro-canonical rate coefficients, in general case, can lead to smaller values of rate coefficient for the process (R1), especially, at low temperatures.83 However, the magnitude of the distinction in the rate coefficients estimated by CVT and µVT even for the reactions with small or close to zero energy barrier mounts to a factor of 1.1-1.3 at T = 300 K.68 Therefore, one can conclude that the drawbacks of the applied methodology for the estimation of rate constant can not be responsible for rather high difference between predictions and measurements. Earlier Starikovskiy and Aleksandrov,23 following to previous studies,26,27 also assumed that the interaction of N2(A3Σu+) with CH4 led to quenching of excited molecule N2(A3Σu+). They recommended the value of rate constant measured by Slanger et al.27 at T = 300 K (k = 1.8×109 cm3mol-1s-1) for the


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whole temperature range typical for combustion conditions. However, as it follows from Figure 5, the rate constant of this process is strongly temperature-dependent. Also, it is worth noting that some researchers supposed that the interaction of N2(A3Σu+) with CH4 resulted in the dissociation of methane molecule (dissociative quenching) N2(A3Σu+) + CH4→ N2(X 1Σg+) + {CH3 + H, CH2 + H2 } with rather high rate constant, which was assumed to be temperature-independent at T = 600– 1200 K.13,20,36 The value of this rate constant was estimated on the basis of comparison with the rate coefficient for the similar reaction of N2(A3Σu+) with other partners, it turned out equal to about 1012 cm3mol-1s-1.36 Analysis of the triplet PES for the N2 + CH4 system exhibited that, indeed, the dissociative quenching of N2(A3Σu+) could proceed, as far as the energy excess Qe, released in the course of the exothermic reaction (R1), is rather high (Qe = 1.78 eV), and it can release mainly into the vibrational degrees of freedom of reaction products: N2H and CH3. Because the N2H dissociation energy barrier is small (De = 0.43 eV),84 the fast decay of vibrationally excited N2H molecule can occur. In order to estimate the rate of this process, let us assume, following to the previous works,85,86 that the energy Qe is distributed uniformly over all vibrational modes of product molecules (3 modes of N2H and 6 modes of CH3). In this case, the energy, released into the vibrations of N2H molecule, Evib, mounts to 0.59 eV. As far as Evib > De, vibrationally excited N2H(V) molecule undergoes fast decay. The rate of this decay, kdec, was estimated on the basis of RRKM theory with the use of the ChemRate program package.87,88 For Evib = 0.59 eV, the computations gave kdec= 1.5×1013 s-1. This value is significantly greater than the Lennard-Jones collision frequency, Zcol, for the N2H(V) molecule with the particle of bath gas M (for M = Ar, at normal conditions, the estimates gave Zcol ≈ 5.4 ×109 s-1). Therefore, we are led to infer that the decay of N2H(V) molecule must occur much faster than its collisional stabilization, and the reaction path (R1) can proceed, as it was assumed earlier,13,20,36 with the formation of N2(X 1Σg+), CH3 and H products, N2(A3Σu+) + CH4 → N2H(V) + CH3(V) → N2(X 1Σg+) + H + CH3.


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As it is seen from Figure 5a, the calculations of the present work for the rate constant of channel (R1) agree rather well with the estimations of Diamy et al. 36 for the dissociative quenching rate coefficient. As to the reaction of N2(A3Σu+) with C2H6 (process (R2)), it was demonstrated that the energy barrier for this reaction channel is smaller than that for the channel (R1). In fact, only CCSD(T) calculations predicted the positive value of Ea for this process (Ea = 120 K), whereas the other quantum chemical methods, including composite ones, gave the negative magnitude of Ea. The rate constants for the process (R2), estimated by using different Ea values, are depicted in Figure 5. As is seen, the uncertainty, associated with the inaccuracy in the calculation of Ea value, is rather small (at T = 300 K, its magnitude mounts to a factor of 1.5). Until now it has been believed that the reaction of N2(A3Σu+) with C2H6, just like the reaction of N2(A3Σu+) with CH4, resulted in collisional or dissociative quenching of N2(A3Σu+). For the reaction of N2(A3Σu+) with C2H6, the formation of C2H5 and H in the course of quenching process is expected. Experimental data on the collisional quenching of N2(A3Σu+) by C2H6 were reported previously8,27 for low temperatures. In these experiments, only the rate of N2(A3Σu+) decay was detected. This means that the possible channels of the reaction of N2(A3Σu+) with C2H6 were not identified in these works. The rate constant of N2(A3Σu+) decay turned out equal to 1.3×1011 – 1.6×1011 cm3mol-1s-1 at T = 300 K.8,27 As far as the reactive channel (R2) prevails over possible collisional quenching, the data reported elsewhere8,27 can be attributed to the rate constant of the process (R2). As it is seen from Figure 5, the rate constant for the process (R2), calculated in the present work for Ea=0 K, is by a factor of 2-5 larger than that reported in the past.8,27 It should be emphasized that earlier Moreau et al.21 assumed that the following dissociative quenching process N2(A3Σu+) + C2H6→ N2(X 1Σg+) + C2H4 + H2 can take place. Moreau et al. also supposed that the rate constant, reported by Slanger et al.27 for collisional quenching, could be attributed to this process. However, as it follows from our analysis, such dissociation channel is hardly probable. At the same time, analogously to the process (R1), the reaction 16 ACS Paragon Plus Environment

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path (R2) also can lead to the formation of vibrationally excited molecules: N2H(V) and C2H5(V). In general, the decay of vibrationally excited N2H(V) molecule can occur, i.e. the interaction of N2(A3Σu+) with C2H6 may be a complex process: N2(A3Σu+) + C2H6 → N2H(V) + C2H5(V) → N2(X 1Σg+) + H + C2H5. In order to estimate the probability of this complex process, let us assume, analogously to the reaction of N2(A3Σu+) with CH4, that the energy, released in the course of the process (R2) (Qe = 1.97 eV), is distributed uniformly over vibrational modes of product molecules (3 modes of N2H and 15 modes of C2H5). In doing so, one can obtain that the energy, coming into N2H molecule vibrations (Evib = 0.33 eV), is smaller than the N2H dissociation energy (De = 0.43 eV). Therefore, the fast decay of excited N2H(V) molecule with the production of N2(X 1Σg+) and H can not occur. Thus, in this case, the occurrence of dissociative quenching is less probable compared to the N2(A3Σu+) + CH4 system. It should be emphasized that the triplet pair of radicals, N2H and CH3, formed in the course of the reaction of N2(A3Σu+) with CH4, is not reacting one (with the exception of possible formation of triplet CH3N2H complex). The fast barrierless reaction (R3) can become spin-allowed, only if the tripletsinglet conversion of radical pair N2H + CH3, induced by magnetic or exchange interactions, occur.89,90 The analogous situation takes place for the reaction of N2(A3Σu+) with C2H6. Therefore, the reactive systems under study involve the spin-selective stages, and their dynamics can be controlled by external magnetic field or by spin-catalytic effect of third spin-carrying particle.89,91 Note that the spin-catalytic effect for these reactions can not be significant in the gas phase at moderate pressures, as far as it comes into play upon triple collisions. Let us discuss now the possible reasons of the difference between our theoretical estimations for the rate constants of processes (R1) and (R2) and measurements.8,26,27 It is worth noting, that in experimental studies26,27 the values of these rate coefficients were obtained by fitting of the measured and calculated time variation of the γ-band radiation of NO molecules that were admixed to the electronically excited nitrogen N2(A3Σu+). In doing so, N2(A3Σu+) molecules were produced via photodissociation of N2O: ACS Paragon Plus Environment

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N2O + hν → N2(A3Σu+) + O(3P). Slanger et al.26,27 proposed some kinetic model involving the following collisional and radiative processes: N2(A3Σu+) + N2O → N2(X 1Σg+, V) + N2O(V) N2(A3Σu+) → N2(X 1Σg+) + hν (Vegard-Kaplan bands) N2(A3Σu+) + NO → N2(X 1Σg+, V) + NO(V) N2(A3Σu+) + NO → N2(X 1Σg+) + NO(A2Σ+) NO(A2Σ+) → NO + hν (γ-bands) NO(A2Σ+) + NO → NO(V) + NO(V) NO(A2Σ+) + N2O → NO(V) + N2O(V) N2(A3Σu+) + M → N2(X 1Σg+, V) + M (M = CH4, C2H6) As is seen, this kinetic model treats a number of processes resulting in the formation of vibrationally excited molecules. As is known, vibrationally excited molecules react much faster than non-excited ones.6,85 However, Slanger et al.26,27 neglected the backward processes with vibrationally excited particles. This can lead to underestimation of real values of rate coefficients. Moreover, upon fitting procedure, it was assumed that admixed CH4 and C2H6 molecules can participate only in quenching of N2(A3Σu+), that is not valid. Note that the relative magnitude of the discrepancy in the predicted rate coefficients compared to measured ones for the reaction of N2(A3Σu+) with C2H6 is smaller than that for the reaction of N2(A3Σu+) with CH4. This may be caused by the fact that the rate constant of the process (R1) is by a factor of ~30 greater than that for the process (R2), and, therefore, the relative contribution of processes, ignored during complex fitting,26,27 is smaller in this case. In order to obtain the temperature approximations for the rate constants of reactions (R1) and (R2), the CVT with the Ea values, estimated by composite methods (Ea = 1015 K for channel (R1) and Ea = 0 K for channel (R2)), was applied for the temperature range T = 200-3000 K. As to the barrierless

reaction channels (R3) and (R4), the corresponding rate constants were estimated in line with the eqs. (1) and (2). In doing so, the following calculated values of dipole moment µ and static polarizability 18 ACS Paragon Plus Environment

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α were taken for the estimations: µ(N2H) = 1.93 D, µ(CH3) = 0 D, µ(C2H5) = 0.29 D, α(N2H) = 2.37 Å3, α(CH3) = 2.29 Å3, α(C2H5) = 4.01 Å3. The electronic degeneracies for the ground state N2H, CH3 and

C2H5 molecules were taken equal to 2. For the reactive PESs, they were taken equal to 1. The obtained temperature-dependent rate constants k(T) for the channels (R1)–(R4) were approximated in line with Arrhenius formula k(T)=AT nexp(-Ea/T) (cm3mol-1s-1). The corresponding coefficients are listed in Table 2. As it follows from our quantum-chemical analysis, the reactions N2H + CH3 and N2H + C2H5 can occur not only through the channels (R3) and (R4), but also through the paths (R5) and (R6). The rate constants of (R5) and (R6) pathways are pressure-dependent. It is worth noting that the existence of barrierless reaction path (R5) was discussed earlier.66,92 In order to estimate the appropriate pressuredependent rate constants, the RRKM-based master equation analysis with the use of the ChemRate program package87,88 was performed for the CH3 + N2H + M → CH3N2H(1A) + M and C2H5 + N2H + M → C2H5N2H(1A) + M reactions. In doing so, the exponential-down model of collisional energy transfer was applied, and the parameter α, governing the magnitude of average energy transferred during collisional deactivation, was specified in a temperature-dependent form. For M = N2, the α values for the CH3N2H(1A) and C2H5N2H(1A) complexes were taken the same as those, recommended for the unimolecular reactions of CH4 (α(T) = 0.25T + 67 cm-1)93 and C2H6 (α(T) = 0.46T + 140 cm-1),94 respectively. The collisional diameter σ and the well depth of Lennard-Jones potential ε for the CH3N2H(1A) and C2H5N2H(1A) molecules were estimated in line with the methodology of Sharipov et al.95 The calculations gave σ = 4.8 Å, ε = 79 K for the CH3N2H molecule and σ = 5.32 Å, ε = 292 K for the C2H5N2H molecule. Figure 6 depicts the temperature dependence for the rate coefficients of the reaction pathways (R3) and (R4) as well as for the reaction pathways (R5) and (R6) for different pressure values. From the plots depicted in figure 6 one can conclude that the rate constants kR3 and kR4 are slightly temperaturedependent (k(T)~T 1/6). It is also seen that their values are very close to each other and are in the range of 2×1013-3×1013 cm3mol-1s-1. Note that the rate constant for the reaction channel (R3), obtained in the 19 ACS Paragon Plus Environment


present work, is very close to that applied in the reaction mechanisms of methane combustion92,96 (k(T)=2.5×1013 cm3mol-1s-1). The rate constants of the formation of CH3N2H(1A) and C2H5N2H(1A) complexes at moderate pressure range (P = 1-50 atm) is substantially smaller than the rate coefficents of (R3) and (R4) reaction channels. So, even at P = 50 atm, the pressure-dependent rate constant of the process (R5) is more than by a factor of 50 smaller than the rate constant of the pathway (R3). The rate constant for the process (R6) is even smaller compared to that for the process (R5). Thus, one can conclude that the probability of the arise of CH3N2H(1A) and C2H5N2H(1A) species is practically negligible in the N2(A3Σu+) +CH4 and N2(A3Σu+) +C2H6 at the pressures P < 50 atm. 14

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Figure 6. Temperature-dependent rate constants k(T) for the reaction channels (R3) and (R4), obtained in the present work (solid curves) as well as the rate coefficients applied for the channel (R3) in the reaction mechanism GRI-3.096 and in the analysis of Tomeczek and Gradon92 (dashed curve). The reaction rate constants for the channels (R5) and (R6) at different pressure values (P = 1 and 50 atm) are depicted by dotted curves.

4. Conclusions Quantum chemical calculations were conducted to examine the potential energy surfaces and to study the reaction kinetics in the N2(A3Σu+) + CH4 and N2(A3Σu+) + C2H6 systems. During the PESs investigations with the usage of the Becke’s hybrid half-and-half density functional BH&HLYP, it was revealed that the interaction of N2(A3Σu+) molecule with CH4 (C2H6) occurred through the barrierless HACS Paragon Plus Environment

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abstraction process that resulted in the formation of N2H molecule and methyl (ethyl) radical. The refinement of energy profiles for these reaction paths by using high-level post-Hartree-Fock and composite methods confirmed that the considered processes possess very small or even zero activation energy. The appropriate rate constants for the N2(A3Σu+) + CH4 and N2(A3Σu+) + C2H6 reaction paths were determined on the basis of variational transition-state theory taking into account tunnel effect. It was found that the rate constant for the N2(A3Σu+) + C2H6 reaction is considerably greater than that for the N2(A3Σu+) + CH4 one. So, at T = 300 K, the ratio of these rate constants is as large as 30. Further, the products of these reaction paths can transform to the ground state N2(X 1Σg+) and CH4 (C2H6) molecules via the degenerated PESs as following: N2H + CH3 → N2(X 1Σg+) + CH4 and N2H + C2H5 → N2(X 1Σg+) + C2H6. At the same time, the possibility of direct collisional quenching of N2(A3Σu+) by CH4 and C2H6 molecules via the PES intersections was concluded to be negligible as compared to the reaction channels. The estimations, based on RRKM analysis, exhibited that the vibrationally excited N2H(V) molecule, formed in the course of the reaction of N2(A3Σu+) with CH4, might dissociate into the N2 and H products much faster than it could be stabilized by collisions with particles of bath gas. Therefore, the reaction of N2(A3Σu+) with CH4 can lead to the dissociative quenching of N2(A3Σu+), and the following elementary reaction can occur: N2(A3Σu+) + CH4 → N2(X 1Σg+) + H + CH3. However, in the case of the reaction of N2(A3Σu+) with C2H6, the fast decay of excited N2H(V) molecule is little probable, and the following process preferably takes place: N2(A3Σu+) + C2H6 → N2H + C2H5. The rate constants of N2H + CH3 → N2(X 1Σg+) + CH4 and N2H + C2H5 → N2(X 1Σg+) + C2H6 processes were determined on the basis of capture model. It turned out that the rate constants of these processes have weak temperature dependence, and their values coincide well with those suggested by other researchers. The RRKM analysis showed that the formation of CH3N2H and C2H5N2H complexes upon the interaction of N2H with CH3 and C2H5 molecules is also possible. However, these channels can be ignored even at rather high pressure (P = 50 atm) because of their small rate coefficients.


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Supporting Information Table comprising the geometrical structure and properties of computed critical points. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgments This work was supported by by Russian Science Foundation (project no. 16-19-00111) and by the Russian Foundation for Basic Research (grant no. 14-08-00794) in the part of the analysis of rate constant for the N2(A3Σu+) + CH4 reaction channel.

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12. Kirillov, A. S. Electronically Excited Molecular Nitrogen and Molecular Oxygen in the HighLatitude Upper Atmosphere. Ann. Geophys. 2008, 26, 1159–1169. 13. Pintassilgo, C.; Loureiro, J. Kinetic Study of a N2–CH4 Afterglow Plasma for Production of NContaining Hydrocarbon Species of Titan’s Atmosphere. Adv. Space Res. 2010, 46, 657–671. 14. Kirillov, A. S. Influence of Electronically Excited N2 and O2 on Vibrational Kinetics of These Molecules in the Lower Thermosphere and Mesosphere during Auroral Electron Precipitation. J. Atmos. Sol. Terr. Phys. 2012, 81–82, 9–19.

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16. Starik, A. M.; Titova, N. S.; Arsentiev, I. V. Comprehensive Analysis of the Effect of Atomic and Molecular Metastable State Excitation on Air Plasma Composition behind Strong Shock Waves. Plasma Sources Sci. Technol. 2010, 19, 015007. 17. Kadochnikov, I. N.; Loukhovitski, B. I.; Starik, A. M. Thermally Nonequilibrium Effects in Shock-induced Nitrogen Plasma: Modelling Study. Plasma Sources Sci. Technol. 2013, 22, 035013(14pp). 18. Guerra, V.; Sa, P. A.; Loureiro, J. Role Played by the N2(A3Σu+) Metastable in Stationary N2 and N2–O2 Discharges. J. Phys. D: Appl. Phys. 2001, 34, 1745.


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Table 1. Values of Energy Barriers Ea for the Processes (R1) and (R2) Calculated at Different Levels of Theory. Ea , K

Method

reaction (R1)

(R2)

CCSD(T)/aug-cc-pvDZ

770 (970) a

120 (230)

QCISD(T)/aug-cc-pvDZ

650 (850)

-70 (45)

G2MP2

655

-1315

G3MP2

960

-1040

G4MP2

1015

–

XMCQDPT2/aug-cc-pvTZ 4500 (4700) a

4055 (4170)

The values of Ea without ZPE correction are given in brackets.

Table 2. Coefficients of Arrhenius Approximation Recommended for the Rate Constants of Reactions under Study. Reaction N2(A3Σu+) + CH4 → N2H + CH3 (R1) → N2(X

1

Σg+)

+ H + CH3 (R1′)

N2(A3Σu+) + C2H6 → N2H + C2H5 (R2) 1

+

A

n

Ea , K

1.82×107

1.823

993.0

6.86×106

1.973

-80.2

12

N2H + CH3 → N2(X Σg ) + CH4 (R3)

7.61×10

0.167

0.0

N2H + C2H5 → N2(X 1Σg+) + C2H6 (R4)

7.70×1012 0.165

0.0


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TOC 3 +

N2(A Σu ) + CH4 3 +

N2(A Σu ) + C2H6 1 +

N2(X Σg )+ CH4

N2H + CH3 N2H + C2H5

1 +

N2(X Σg )+C2H6


32

Theoretical Study of the Reactions of Methane and Ethane with

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