The O + NO(v) Vibrational Relaxation Processes Revisited - The

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The O+NO(v) Vibrational Relaxation Processes Revisited Pedro J.S.B. Caridade, Jing Li, Vinícius C. Mota, and António J.C. Varandas J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03431 • Publication Date (Web): 24 May 2018 Downloaded from http://pubs.acs.org on May 24, 2018

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The Journal of Physical Chemistry

The O+NO(v) Vibrational Relaxation Processes Revisited P. J. S. B. Caridade,∗,†,‡ Jing Li,¶ V. C. Mota,§ and A. J. C. Varandas∗,¶,∥ †Chemistry Centre and Chemistry Department, University of Coimbra, 3004-535 Coimbra, Portugal ‡Institute for Interdisciplinary Research, University of Coimbra, 3004-535 Coimbra, Portugal ¶School of Physics and Physical Engineering, Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Qufu Normal University, 273165, Qufu, China §Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES 29075-910, Brazil ∥Chemistry Centre and Chemistry Department, University of Coimbra, 3004-535 Coimbra, Portugal E-mail: [email protected]; [email protected]

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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

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Abstract We have carried out a quasiclassical trajectory study of the O + NO(v) energy transfer process using DMBE potential energy surfaces for the ground-states of the 2 A′

and 2 A′′ manifolds. State-to-state vibrational relaxation rate constants have been

computed over the temperature range 298 K and 3000 K and initial vibrational states between v = 1 and 9. The momentum-Gaussian binning approach has been employed to calculate the probability of the vibrational transitions. A comparison of the calculated state-to-state rate coefficients with the results from experimental studies and previous theoretical calculations shows the relevance of the 1 2 A′′ potential energy surface to the title vibrational relaxation process.

1 Introduction The atmosphere cooling mechanism is mainly dictated by radiative molecular emission in the middle infrared (IR) region. 1 In the case of the Earth atmosphere, the dominant species N2 and O2 are infrared inactive and hence do not irradiate in the 2 − 15 µm region, while minor constituents such as nitric oxide and carbon dioxide are important sources in the IR emission background of the thermosphere, 2–7 which extends from about 90 km to between 500 and 1, 000 km above our planet with temperatures that can range from about 500 o C (773 K) to 2, 000 o C (2, 273 K) or higher. The cooling mechanism is initiated by one-quantum excitation in collisions with atomic oxygen followed by spontaneous emission: 8,9 NO(v = 0) + O + 1875 cm−1 → NO(v = 1) + O

(1)

NO(v = 1) → NO(v = 0) + 5.3 µm

(2)

At 100 km of altitude, collisional quenching and spontaneous emission become of the same order of magnitude, with the latter dominating at higher altitudes. Other atmospheric species do not contribute significantly to the energy transfer processes even though their concentra2


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tions are significantly higher than atomic oxygen. For example, vibrational excitation via collisions with O2 and N2 are 103 − 105 less effective 10–13 despite the fact that the mixing ratios with atomic oxygen are ∼ 10−2 at 100 km. Rather than measuring the one quantum excitation of reaction (1), experimental and theoretical studies have focused on the de-excitation process using the micro-reversibility hypothesis to calculate the rate constant for the direct reaction. In fact, early work by Kaplan et al. 14 has shown that such a reversibility condition may be applicable to vibrational excitation if the reactants are thermalized. On the other hand, a quasiclassical dynamics study 15 of the excitation process O + NO(v = 0 → 1) concluded that such a method can be statistically challenging and the micro-reversibility process applicable only if the translational and rotational thermal conditions are verified both for the reactants and products. 15 Experimental work by Hwang 16 has additionally shown that such an assumption is valid. In turn, by invoking non-local thermodynamic equilibrium, a decrease in collision frequency may lead to hotter or colder internal degrees of freedom than the translational temperature. Having in mind that other surrounding species may be in thermal equilibrium, experimental and theoretical vibrational relaxation should therefore cover a wide temperature range. Experimentally, several studies have been published along the years focusing on the O + NO(v) vibrational relaxation, with large discrepancies and uncertainties being reported. The reference data for the one-quantum vibrational relaxation rate constant at room temperature, kv=1→0 (T = 298 K) = 6.5 × 10−11 cm3 s−1 , comes from the work of Fernando and Smith 10 and has been utilized to interpret the radiance data of the lower thermosphere. Using the laser pump-probe technique, Dodd et al. 17–19 studied the temperature dependence of the vibrational relaxation for NO(v = 1, 2, 3). A fundamental conclusion of this series of publications is that kv=1→0 (T = 298 K) = 2.4 ± 0.5 × 10−11 cm3 s−1 , thence nearly three times smaller that the results of Fernando and Smith. 10 The use of such data in the Global Mean model to describe the Earths’ thermosphere leads to unrealistic temperature and density structure profiles, although this behavior could be minimized by employing a removal rate

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constant three-times larger. In 2003, a new experimental study has been reported by Dodd’s group, using a new source of O(3 P ) for the same laser pumping experiments of Ref. 16. From this, a 75% larger removal rate constant has been reported for the v = 1 → 0 transition, thus becoming consistent with other experimental data and macro-kinetics simulations. From the theoretical point of view, two dynamical studies have been reported in the literature, both using quasiclassical trajectories (QCT). Duff and Sharma 20 presented an adiabatic study using the 12 A′ and 12 A′′ states, which are known to play a significant role in the energy transfer process. Their many-body expansion (MBE) functions 21 have been calibrated using ab initio data 22 that covered regions near the NO2 C2v minimum. To elucidate the dynamics of the title reaction, a detailed QCT study has also been reported by our group 23 using an accurate multi-sheeted double-many-body-expansion (DMBE) potential energy surface 24 for the 2 A′ manifold. For the 12 A′′ state, the MBE potential surface of González et al. 25 has been utilized but after being modified to describe approximately the O + NO(2 Π) long-ranges forces. Both dynamics studies 20,23 and other theoretical work reported in the literature 26,27 show very good agreement with the experimental results of 1997, 19 i.e., are unable to reproduce the most recent results 16 or recommended data. 10 Since the 2 A′ DMBE potential energy surface for NO2 is known to give accurate dynamical results 28 and reproduces the crossings and avoided-crossings of the manifold, 24 the most probable issue for the experimental-theoretical discrepancies is the 2 A′′ function. In fact, a recent analysis 29 has shown that the potential energy surface is topographically intricate requiring a diabatic potential matrix of at least 3 × 3 dimensionality, which is not a trivial task. In turn, the González et al. 25 function is based on the MBE formalism 21 with the calibrating procedure based on perturbation theory/full-valence complete-active space method 30 (CASPT2) and the correlated consistent polarized triple-zeta (VTZ) basis set 31 data. Using the switching-function scheme of Varandas and Poveda, 32 an accurate DMBE global single-sheeted surface has been reported 33 based on internally contracted multi-reference configuration interaction 34,35 (MRCI) and the augmented-VQZ (AVQZ) ba-

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sis set. 31,36 Several topological features have been highlighted in this study, in particular the relative position of the global 2 B1 minimum at linear geometries which connects with the one of Cs symmetry 25 ) by a barrier of 9.7 kcal mol−1 . The goal of this paper is to use the novel DMBE function for the 2 A′′ state in revisiting the O+NO(v) vibrational relaxation process with the QCT method while employing the momentum Gaussian binning (MGB) scheme rather than the usual semi-classical quantization (SCQ) techniques for analyzing the trajectories. Such a study will cover vibrational states up to v = 10 and temperatures ranging from 298 K up to 3000 K. In order to provide a direct comparison with the state-to-all experiments of Hwang et al., 16 additional calculations have also been performed for the 2 A′ state. After briefly describing the most relevant attributes of the 2 A′′ DMBE surface in section 2, the dynamical methods are presented in section 3. The focus will then be on the initial conditions of the reactant diatomic molecules and the final vibrational rotational quantization. The results are presented and discussed in section 4, and the conclusions gathered in section 5.

2 Potential energy surfaces The O(3 P ) + NO(2 Π) channel correlates with six states of

2,4

A′ and

2,4

A′′ symmetry if spin-

orbit coupling is neglected. Although a total of up to 36 states can be involved in the dynamical process, only two states of the doublet ansatz, 2 A′ and 2 A′′ , have been considered due to the presence of stable minima, since they allow a more effective randomization of the internal energy (see Ref. 23). Figure 1 shows the relative position of the two states with stable minima that play a relevant role in the vibrational relaxation dynamics. Note however that the remaining repulsive states may still have a significant role in the energy transfer process, especially for high translational energies. As in previous work, the DMBE/ES 8 × 8 matrix is used to describe the 2 A′ manifold, being the details of the topography of the potential energy surface there reported and in the original paper. 24 For the 2 A′′ state, the

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0

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

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O+NO(v’) → O+NO(v’’)

O(3P)+NO(2Π)

−20 2

C2v (TS)

A’’ (Cs)

−40

2

Πu

−60

−80

NO2(2A1)

reaction coordinate

Figure 1: Energy diagram showing the two electronic states and their main attributes arising from the DMBE PES for 24 1 2 A′ and 33 1 2 A′′ electronic states of NO2 . single-sheeted global ab initio-based DMBE potential energy surface 33 has been employed, which is briefly described in the remain of the section. The lowest state of the 2 A′′ manifold dissociates according to the scheme: 29,33 NO(2 A′′ ) → NO(2 Π) + O(3 P )

(3)

2 → O2 (3 Σ− g ) + N( D)

(4)

→ O2 (3 ∆u ) + N(4 S)

(5)

where the two last channels depend of the O2 internuclear distance: dissociation via (4) if ROO ≤ 2.8 a0 , by channel (5) otherwise, smoothly switching between both regimes. The presence of the N(4 S)/N(2 D) electronic states in the dissociation schemes (4) and (5) forces the use of either a diabatic potential energy matrix formalism or to approximate such behavior by means of a switching function scheme. Although the former is physically motivated, the mapping of the potential energy surface using a switching function is known to give accurate results, except in what concerns the vicinity of the electronic crossing seam. 6


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This is one of the most relevant features of the DMBE NO2 (12 A′′ ) PES, 33 and appears as a consequence of an asymptotically evolving crossing seam modeled via a single-sheeted formalism. As depicted elsewhere, 29 for ROO ≤ 2.8 a0 the lower adiabat merges asymptotically 2 into the NO2 (12 A2 ) diabat yielding O2 (3 Σ− g ) + N( D), while for ROO > 2.8 a0 the merging

occurs with the NO2 (12 B1 ) diabat, which correlates with O2 (3 ∆u ) + N(4 S). A reliable modeling of the complete ROO interval with a single-valued form requires the use a switching function formalism. For the NO2 (12 A′′ ), this leads to a DMBE form written as 33 (1)

(2)

(2)

(2)

V (R) = VN(4 S)/N(2 D) f (R) + VOO (R1 ) + VNO (R2 ) + VNO (R3 ) + V (3) (R)

(6)

(1)

where VN(2 D)/N(4 S) is pseudo-one-body atomic energy splitting form, which is parameterized 2 to interconnect the O2 (3 Σ− g ) + N( D) and ROO > 2.8 a0 regimes via a multiplicative f (R)

switching function. In turn, V (2) and V (3) are two-body and three-body energy components (respectively) which, within the DMBE formalism, are decomposed into extended-HartreeFock (EHF) and dynamical-correlation (dc) contributions. As usual in DMBE theory, the two-body energy terms correctly describe themselves the asymptotic energy contributions, while the three-body ones take care of the contributions due to the atomic quadrupolediatomic dipole and quadrupole long-range electrostatic energy contributions. The data used for the fit of the functional form is based on three-state averaged CASSCF, followed by one-state MRCI calculations using the AVQZ basis set, which cover the full configuration space. Indeed, the DMBE form is based on a fit to 1681 MRCI energies once scaled a posteriori using the DMBE-SEC 37 method (thence, by scaling their dynamical or external correlation energy component) to account for excitations beyond singles and doubles and correct for the incompleteness of the basis set. The overall root mean-squared deviation gives chemical accuracy, less then 1 kcal mol−1 for up to 1000 kcal mol−1 above the absolute minimum, being of ∼ 0.6 kcal mol−1 over most of the relevant configurational space. Note however, that for such extremely high values of the interaction energy, such a value is merely

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qualitative since crossings and avoided-crossings with other states are surely present. Two variants of the PES have been presented by focusing on distinct approaches to the 2 A2 /2 B1 crossing regions. In fact, as discussed elsewhere, 29,33 the 2 A2 /2 B1 crossing seam separates the lowest energy region of the PES containing the linear 2 B1 global minimum from the region containing the distorted Cs local minimum and several other stationary states. However, due to the single-valued nature of the DMBE functional form, such a crossing can only be approximately described, showing itself as a pseudo transition state. This is the essence of the first form, denoted as DMBE PES, which describes only roughly the region of the 2 A2 /2 B1 crossing seam. To further improve this region, a second variant named DMBE+G PES has also been proposed by adding damped gaussian to the first DMBE form with the solely purpose of accurately describing a set of ab initio energies covering the 2 A2 /2 B1 crossing regions while preserving all other attributes of the global fit. As a result, the region containing all stationary states of the PES, thus including the portion of the 2 A2 /2 B1 crossing region nearby the the linear 2 B1 , is described by the DMBE+G PES variant with a RMSD below ∼ 0.9 kcal mol−1 . In this work, we employ the DMBE+G PES since it provides a good description of the transition state that arises from the adiabatic treatment of the 2 A2 /2 B1 crossing (TS5 in the notation of Ref. 33), which is expected to play a key role on the studied dynamics process. Figure 2 shows a global view of its most relevant features via a relaxed contour plot 38 in hyperspheric coordinates:

β⋆ =

√

3(R22 − R32 )/Q

γ ⋆ = (2R12 − R22 − R32 )/Q

(7) (8)

where Q = (R12 + R22 + R32 ), and R1 is the OO distance. The most relevant feature of the potential energy surface is the global minimum corresponding to the linear 2 Πu structure (β ⋆ = 0, γ ⋆ = 1), which is absent in the MBE form of González et al. 25 and arises from the

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-0.11

2

1.0

NO2(1 A’’)

-0.13 -0.15

0.5

0.0 -0.21

energy/Eh

-0.17 -0.19

γ*

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


-0.23

-0.5

-0.25 -0.27

-1.0

-0.29 -1.0

-0.5

0.0

0.5

1.0

β* Figure 2: Relaxed contour plot 38 of DMBE+G PES 33 for the 1 2 A′′ electronic state of NO2 . 2

A2 /2 B1 crossing located at (β ⋆ = ±0.091, γ ⋆ = 0.890). In turn, as referred to above, the

linear global minimum is connected to the Cs distorted structure (β ⋆ = ±0.154, γ ⋆ = 0.711) via TS5 with a barrier height of 32.25 kcal mol−1 ; the reader is addressed elsewhere, 33 where a detailed description of the DMBE+G PES topographical features can be found.

3 Computational details: quasiclassical trajectories The QCT method 39,40 and strategy used in the present dynamics study of the O + NO(v) vibrational relaxation process are the same as used elsewhere, 23 and hence only the most relevant features are here surveyed. The state-to-state thermal rate constant can be written for the initial vibrational state of NO(v ′ ) as 2 ′

2 ′′

kv′ →v′′ (T ) = kvA′ →v′′ (T ) + kvA′ →v′′ (T )

9


(9)


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with ( kvx′ →v′′ (T )

= ge (T )

8kB T πµO+NO

)1/2 (

1 kB T

)2 ∑

′ (2j + 1) exp(−Ev′ j /kB T )Q−1 j (T ; v )

j

∫ × 0

∞

Etr σvx′ →v′′ (Etr , j) exp

( ) Etr − dEtr (10) kB T

where ge (T ) = 2/{[5 + 3 exp(−227.8/T ) + exp(−326.6/T )][2 + 2 exp(−177.1/T )]} is the electronic degeneracy factor, kB the Boltzmann constant, µO+NO the reduced mass of reactants, Evj the rovibrational energy of the state (v, j), Qj the rotational partition function for the x 23 the v state, and σv→v ′ the vibrational state-to-state cross section. As previously noted, 2

A′ and 2 A′′ states contribute equally for the calculation of the rate constant, with the

trajectories carried out independently for each of the two indicated state symmetries. The integral on Eq. (10) can be effectively evaluated employing the Monte-Carlo method. 39 For the translation energy sampling, and although a exact solution can be found elsewhere 41 by means of numerical root-finding, a simpler approach may be employed using

Etr = −kB T ln(ξi ξj )

(11)

where ξi and ξj are two independent freshly generated random numbers for each trajectory. The rotational state is also obtained by Monte-Carlo sampling assuming a Boltzmann distribution, for which is used the cumulative function

C(j) =

j ∑

(2j + 1) exp(−Evj )Q−1 j (T ; v)

(12)

jmin

being the summation performed until the condition C(j) ≥ ξ is verified. The corresponding rovibrational energy Evj is calculated from the numerical solution of Schrödringer equation 42 for the NO(2 Π) assuming jmin = 1. The use of the Monte-Carlo sampling for the translational

10


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energy and rotational quantum number simplifies Eq. (10) to read ( x kv→v ′ (T )

= ge (T )

8kB T πµO+NO

)1/2 x σv→v ′ (T )

(13)

The study here presented covers initial states v = 1 − 10 and the following temperatures T = 298 K, 530 K, 640 K, 825 K and 1500 K, thus allowing a direct comparison with the experimental results. 16 Since in previous work 23 some of these temperatures had not been considered, additional calculations had to done for the 2 A′ state. For each (v ′ , T ) combination, 5000 trajectories have been integrated on 2 A′′ state using a time-step of 2×10−17 s. Such a number of trajectories is sufficient to minimize statistical fluctuations of the state-to-all rate constant, warranting a convergence up to 1 − 2 %. The maximum impact parameter has been determined a priori by integrating a small set trajectories for fixed values and choosing the smallest value for which the internal energy transfer was negligible during the collision process. Within the QCT framework, the cross section can be replaced by

σvx′ →v′′ (T ) = πb2max Pvx′ →v′′ (T )

(14)

with Pvx′ →v′′ = Nvx′ →v′′ /N the state-to-state probability, Nvx′ →v′′ the number of trajectories that undergoes v ′ → v ′′ transition in a set of N trajectories for a temperature T . For the state-to-all, one gets

Pvx′ ,dex (T ) =

∑

Pvx′ →v′′ (T )

(15)

kvx′ →v′′ (T )

(16)

v ′′

The O + NO(v) Vibrational Relaxation Processes Revisited - The

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