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Closer Versus Long Range Interaction Effects on the NonArrhenius Behaviour of Quasi-Resonant O + N Collisions 2

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Ernesto Garcia, Alexander K. Kurnosov, Mario Antonio Cacciatore, Fernando Pirani, and Antonio Lagana J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 09 Jun 2017 Downloaded from http://pubs.acs.org on June 10, 2017

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The Journal of Physical Chemistry

Closer versus Long Range Interaction Eects on the non-Arrhenius Behaviour of Quasi-Resonant O2 + N2 Collisions A. Kurnosov,† M. Cacciatore,‡ F. Pirani,¶ ‡ A. Laganà,¶ C. Martí,¶ and E. ,

Garcia∗ § ,

†Troitsk

Institute of Innovation and Fusion Research, 142092 Troitsk, Moscow, Russia ‡Nanotec-Institute for Nanotechnology, CNR, Via Amendola 122/D, 70126 Bari, Italy ¶Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, 06123 Perugia, Italy §Departamento de Quimica Fisica, Universidad del Pais Vasco (UPV/EHU), 01006 Vitoria, Spain E-mail: [email protected]

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Abstract

We report in this paper an investigation on energy transfer processes from vibration to vibration and/or translation in thermal and subthermal regimes for the O + N system performed using Quantum-Classical calculations on dierent empirical, semiempirical and ab initio potential energy surfaces. In particular, the paper focuses on the rationalization of the non-Arrhenius behaviour (inversion of the temperature dependence) of the quasi-resonant vibration-to-vibration energy transfer transition rate coecients at threshold. In order to better understand the microscopic nature of the involved processes, we pushed the calculations to the detail of the related cross sections and we analysed the impact of the medium and long range components of the interaction on them. Furthermore, the variation with temperature of the dependence of the quasi-resonant rate coecient on the vibrational energy gap between initial and nal vibrational states and the eectiveness of Quantum-Classical calculations to overcome the limitations of the purely classical treatments were also investigated. These treatments, handled in an Open Molecular Science fashion by chaining data and competences of the various laboratories using a grid empowered molecular simulator, have allowed a rationalization of the dependence of the computed rate coecients in terms of the distortion of the O -N conguration during the diatom-diatom collisions. A way of relating such distortions to a smooth and continuous progress variable allowing a proper evolution from both long to closer range formulation of the interaction and from its entrance to exit channel (through the strong interaction region) relaxed graphical representations, is also discussed in the paper. 2

2

2

2

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1 Introduction A key step of the calculation of the eciency of the energy transfer processes performed by running the INTERACTION and DYNAMICS modules of the Grid Empowered Molecular Simulator (GEMS) 13 leveraging on the competences of the members of the Virtual Organization (VO) COMPCHEM 1 and of the Chemistry Molecular and Materials Science and Technologies (CMMST) Virtual Research Community (VRC), 4 is the assemblage of an appropriate Potential Energy Surface (PES). This is of particular importance for gas phase simulations in which not only short but also long range components of the interaction play a key role and therefore a proper description of what happens when collision partners approach each other from far away is important (see for example plasma chemistry, aerothermodynamics, astrochemistry, etc. applications). 512 In principle, the most accurate way of building a PES within the Born-Oppenheimer scheme consists in performing for each involved molecular geometry a high level ab initio calculation of the related electronic structure. This scheme (called also on-the-y when it follows, as in trajectories integration, the evolution of the considered molecular system) couples together in a single step INTERACTION and DYNAMICS calculations. This approach, however, while making the computational scheme simpler, is both quite expensive (for example dicult to parallelize and likely to unnecessarely repeat the calculation for the same molecular geometry) and makes the continuity of the potential energy and of its derivatives dicult to enforce. Alternative to this is the more popular o-the-y scheme consisting of the following steps: a) produce and/or collect from dierent sources high level ab initio information on the electronic structure of the investigated molecular system, b) check the computed ab initio values for continuity and convergence to the desired electronic states, c) t the checked data to (one or more) suitable analytical formulations of the PES, d) code the PES into a highly performing routine for debugging and validation against available experimental and theoretical information, e) running dynamics codes for the production of observable quantities. 3 ACS Paragon Plus Environment


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However, despite the linearity of the procedure outlined above, in the literature this is often the least "open" ingredient of a molecular dynamics calculation and leads to non homogeneous comparisons between results published by dierent authors. In particular, for diatom-diatom systems, as is the title one, diculties arise when trying to formulate both the more structured inner (short range) region of the interaction and its evolution into one (or more) asymptotic channel(s) using the same functional form. This is, indeed, the case of the procedure adopted for the N 4 system (see refs. 1317) and for the title system O2N2 (see the PES of ref. 18 named here MN). The recent advances in High Performance and High Throughput networked computing, have, in fact, boosted the calculation of a large number (several ten thousands) of high level ab initio potential energy values for the dierent molecular geometries. This has made it possible, as done for the MN PES of the O 2N2 system, to t the calculated ab initio values using a permutationally invariant polynomial in mixed exponential-guassian variables enabling the description of all the nine arrangements of the system corresponding to dissociation, the forming of the N 2O and the NO2 triatoms plus the fragmentation into all the possible atom and diatom subsystems. For this reason, when beginning the investigation of the collisional inelastic vibrational energy transfers for the title system, we rst considered the adoption of the MN potential for carrying out the related calculations. However, the quantitative analysis of the state-to-state rate coecients, of the second virial coecient, of the integral non-reactive cross section and of the opacity function computed on the MN PES led us to conclude that the formulation of the MN PES is denitely inadequate to be used for studies of the non-reactive O 2 + N2 energy transfer processes in thermal and subthermal regimes. 19 Furthermore, the detailed graphical analysis of the MN PES conrmed that, despite the large number of molecular geometries considered, the asymptotic tail of the interaction is completely misrepresented and this prevents the reproduction of available experimental Vibration-Translation (V-T) and Vibration-Vibration (V-V) inelastic rate information. As a consequence, in order to carry out the extended investigation of the energy exchange 4 ACS Paragon Plus Environment

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processes in O2 + N2 collisions for a wide range of initial vibrational excitations in thermal and subthermal regimes and, in particular, in order to single out the conditions in which non-Arrhenius behaviours (inversion of temperature dependence) occur for quasi-resonant V-V transitions, in our computations we used both an older empirical PES (GB1 20 ) and two more recently formulated semiempirical PESs (MF1 21 and MF2 22 ) tailored to realistically reproduce the long range interaction. Main details on how these other PESs are formulated in terms of an

intra-

and an inter-

molecular component (expressed either as a sum of the separated diatomic potentials or as a sum of the isotropic and anisotropic terms of the long-range interaction between the two separated diatoms) are provided in Section 2. The outcomes of an extended investigation of the non reactive properties of the title system are given in Section 3 where to the end of singling out the key features of quasi-resonant transitions leading to non-Arrhenius behaviours we analyse both the thermal rate coecients and the xed energy detailed cross sections computed using Quantum-Classical (QC) (called also semi-classical, SC) techniques. 23,24 In Section 4 eorts to further characterize non-Arrhenius behaviours and to investigate when Quasi-Classical Trajectory (QCT) 25 techniques can be used, are commented. In Section 5 a

progress

variable of the Bond Order (BO) type (smoothly evolving from the entrance to

the exit channel) useful to pinpoint small variations in similar semiempirical PESs (though still leading to appreciable dierences in the close interaction region). In Section 6 some conclusions are drawn.

2 The diatom-diatom O2 + N2 PES In this section we summarize the procedure adopted in the INTERACTION module of GEMS in order to deal with the systematic study of the eciency of the V-V and V-T energy transfers in non reactive processes (for further details see refs. 2628 (N 2 + N2 ) and refs. 21,22 (O2 + N2 )).

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In the above mentioned studies the potential energy an

inter

V

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is partitioned into an

intra

component as follows:

V = Vintra + Vinter both formulated using semiempirical expressions. functions (De (1

βdiat

and

− exp[−βdiat (r − re )]) − De

its force constant and

parameters: 5.213 eV,

re

29

for N2

De

re

Vintra De

is expressed as a sum of two Morse

being the diatomic dissociation energy,

its equilibrium distance) using the following values of the

= 9.905 eV,

= 1.208 Å, and

with

(1)

βdiat

re

= 1.098 Å, and

= 2.654 Å

In the oldest formulation (GB1)

20

−1

βdiat

= 2.689 Å

−1

De

; for O2

=

.

of the O2 + N2 PES

Vinter

is formulated as a com-

bination of two types of components: SR (the Short Range component) expressed as sum of exponentials in the rst and second power of the involved the

ij

rij

internuclear distance of

atomic pair and LR (the Long Range component) expressed as a sum of a perma-

nent quadrupole-permanent quadrupole electrostatic interaction plus an average attractive dispersion. In the more recently proposed MF PESs,

21,22

Vinter

is formulated in terms of the isotropic

and anisotropic components of interaction of the two molecules by means of an expansion in spherical harmonics and bond-bond pairwise additive interactions (see refs 27,30). Then in the INTERACTION module of GEMS high level ab initio electronic structure data are either collected from the literature and/or

ad hoc

produced. Finally in the FITTING module ab

initio information and experimental scattering and second virial coecient data are jointly best tted by varying the value of the parameters of the semiempirical formulation of in terms of diatomic bond components of the dipole electronic polarizability, the diatomic electric quadrupole moment,

Q(r),

(with

r

α(r),

Vinter

and of

being, as usual, the intramolecular

distance) as discussed in refs. 21,22. In the simplest MF formulation (MF1) the two molecules are set at equilibrium and

6


Vinter

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is expressed as a combination of the two "eective" interaction components

VvdW

Velect

and

Vinter = VvdW + Velect .

(2)

representing the van der Waals (size or Pauli repulsion plus dispersion attraction) and the electrostatic interaction terms, respectively.

21,31

The

VvdW

component is formulated as a

bond-bond contribution that takes into account indirectly three body like eects.

32

The

Velect

component is instead formulated as an electrostatic interaction associated with an anisotropic

a and b)

distribution of the molecular charges over the two interacting bodies (say molecules

that asymptotically tends to the permanent quadrupole permanent quadrupole interaction. Both a

VvdW

progress

and

Velect

components are allowed to vary during the collision as a function of

variable quantifying the extent of evolution of the process.

R

excluded, the intermolecular distance (let us take

a ≡

O2 and

b ≡

rb ,

respectively, and

Φ

Θa

and

Θb

formed by

VvdW

6 VvdW (R, γ) = ε(γ) n(x) − 6 x

(Θa , Θb , Φ)

and

n(x)

of eq. 3 (

and

R, ra )

and (

ε

and

Rm

n(x) = β + 4.0 x2 ) β

x = R/Rm (γ)

b

Velect and

R, rb ).

n(x) 6 n(x) 1 1 − x n(x) − 6 x

33

(3)

with

γ

denoting

being respectively the well depth of the

interaction potential and the equilibrium value of denition of

VvdW

and

R with the internuclear vectors ra

is the reduced distance of the two bodies dened as

the triplet of angles

variable with

is of the Improved Lennard-Jones (ILJ) type:

where

progress

being the dihedral angle formed by the planes (

The formulation adopted for

a

between the centres of mass of molecule

N2 ) is taken as such

depending on the Jacobi angles

When reaction is

R

at each triplet of

γ

values.

In the

is a parameter depending on the nature

and the hardness of the interacting particles leading to a more realistic representation of both repulsion (rst term in square brackets of eq. 3) and attraction (second term in square brackets of eq. 3) as discussed in refs. 33,34. For the title system typical of neutral-neutral systems,

33

ε and Rm

β

is given the value of 9 as

are expanded in terms of the bipolar spherical

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harmonics AL1 L2 L (γ) and the reduced form of the bond-bond potential 34 is taken to be the same for all the relative orientations (see refs. 3538). In this way the convergence of VvdW is signicantly accelerated 31,3941 and the value of the ε and Rm expansion parameters can be estimated from diatomic (or molecular bond) polarizability (see Appendix A of ref. 31) and then ne tuned by tting experimental data and accurate ab initio electronic structure calculations. 42 In the MF1 PES used for the study reported here (see also refs. 21,22) Velect is formulated as Velect (R, γ) =

Qa Qb 224 A (γ) R5

(4)

by only retaining the main quadrupole-quadrupole contribution. In eq. 4 Qa and Qb are the already mentioned diatomic permanent quadrupole moments evaluated using ab initio calculations (whose r dependence has been accurately tted to a fth order polynomial). As already mentioned, a ne tuning has been carried out for ε and Rm using ab initio calculations of the intermolecular interaction energies as well as the analysis of second virial coecient (available for a wide range of temperatures) and of the integral cross sections data characterized by an oscillatory pattern associated with quantum interference eects. To this end, for ve selected geometries of the interacting O 2 + N2 system (with both diatoms set at their equilibrium distance), SAPT(DFT) (symmetry-adapted perturbation theory) calculations were performed utilizing a density functional description of the monomers and a large basis set. 43 In order to more accurately account for the involvement of closer interaction regions of the potential we consider here a more exible version of the MF PESs (MF2) in which an actual dependence of α(r) and Q(r) on both ra and rb is introduced by formulating Vinter as a true bond-bond interaction. This marks a real dierence between MF1 (whose Vinter parameters of eq. 1 are related to the values of Qa , Qb and α associated with the molecular monomers at equilibrium) and MF2 (whose Vinter parameter values of Qa , Qb and α depending on the value of r and where related values can be found in refs. 21,22,26,27. A graphical analysis 8


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of the dierent shapes of the various cuts of the MF PESs are given in ref. 19 where, as mentioned in the Introduction, the inadequacy of the MN PES to represent the long range tail of the interaction is discussed.

3 Detailed properties of the O2 + N2 collision processes In this section, following the scheme adopted for the already mentioned studies of the N 2 + N2 system and leveraging on the ndings illustrated there, 26,27 we discuss the non reactive dynamical and kinetic properties of the O 2 (v) + N2 (u) → O2 (v ) + N2 (u ) collision processes (where u and v are the initial vibrational states of the colliding molecules and primed quantities are the corresponding nal ones). Before undertaking the planned extended calculations of the state to state rate coecients for non reactive collisions on the above mentioned GB1, MF1 and MF2 PESs, 2022 we checked the suitability of our QC approach to compute thermal quasi-resonant rate coecients for the title system. At present, QC approaches are, in fact, the most accurate techniques of general use to investigate thermal and subthermal four atom collisions due to the fact that no full quantum dynamics calculations can be realistically performed for the fairly heavy atoms and high vibrational levels of the system considered here. However, in order to understand when less accurate approaches can be adopted for the investigation of inelastic collision processes, a comparison with the outcomes of QCT calculations will be discussed in more detail in the next section. A description of the quantum-classical method used and the values of the parameters can be found in refs. 2124,26. A rst comparison between experimental data 44 and computed QC quasi-resonant rate coecients k(v, u|v , u ) for v = 13 and 19 with u = 0 (to v = 11 and 17 and u = 1) is given in Table 1. The calculations were performed on the GB1, MF1 and MF2 PESs at the temperature (300 K) at which usually quasi-resonant rate coecients show peculiar features (as we shall comment in more detail later; see also the extended discussion given on this point in ref 26 for the N 2 + N2 system). From Table 1 it is apparent that the agreement

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with experimental data of the QC results computed on MF2 is excellent while it is less good with those computed on GB1 and MF1. The distinctive feature of the quasi-resonant transitions is the near exact transfer of the vibrational energy of one collision partner into that of the other (ie almost a null total value of the dierence ΔE between the initial vibrational energy content and the nal one ΔE = ΔEv u −vu = Ev + Eu - Ev - Eu with Es being the vibrational energy associated with state s) and the quantum-like nature of the peculiar dependence of the related rate coecients on the temperature T that makes their characterization quite a challenge. We recently found, in fact, that while quasi-resonant N 2 + N2 QCT rate coecient values in general monotonously decrease as T lowers (at low temperatures) this is not always true for the QC ones which, as pinpointed by the experimental work of Martínez et al., 45 show in certain cases a non-Arrhenius behaviour. 26,27 This non-Arrhenius behaviour is what is shown also by the O2 + N2 system whose quasi-resonant detailed rate coecient k(19, 0|17, 1) (that is the short form for indicating the rate coecient k(v = 19, u = 0|v = 17, u = 1) involving the excitation of N2 from the ground vibrational state u = 0 to the rst excited one u = 1 and the simultaneous double deexcitation of O 2 from v = 19 to v = 17 with ΔE = 6.9 meV) computed on MF2 is plotted over an extended range of temperature in Fig. 1 together with those computed on GB1 and MF1. The Figure shows that the k(19, 0|17, 1) values computed on MF2 decrease almost one order of magnitude in going from T = 6000 K to T = 1000 K and then (around 500 K) invert the trend when the temperature lowers further (non-Arrhenius behaviour). On the contrary k(19, 0|17, 1) values computed on both the GB1 and the MF1 PESs decrease more than one order of magnitude when the temperature lowers from 6000 K down to 1000 K and then another order of magnitude when T lowers from 1000 K to room temperature. As a result, at T larger than 2000 K the QC rate coecient values computed on MF2 for the considered transition are not signicantly dierent from those computed on MF1 (suggesting that at high temperature the dierences between the dierent MF potentials are small) while being

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smaller than the values computed on GB1. On the contrary, when

T

further lowers the

rate coecient values computed on the MF potentials decrease less than those computed on GB1 so as to eventually cross them and become larger at temperature values lower than 1000 K. This eect is particularly evident for the MF2 results because after the crossing they even invert the slope and give rise to a non-Arrhenius behaviour. This is due to the mutual increasingly distorting eects of O 2 and N2 monomers associated with the already discussed actual dependence of

α(r)

and

Q(r)

on

r

in MF2 that allows the overall conguration of

the system to adapt to the increased interaction between the two diatoms. In other words, as already found in refs. 26,27,45 for the N 2 -N2 quasi-resonant transitions, closer distance distortions enhance the eect of the long range attractive component of the PES and exalt the role of the shorter range MF2 exibility (as will be further commented later) whose balance controls the related enhancement of non reactive energy transfer mechanisms. In order to rationalize this feature, we pushed the analysis to a more detailed level by computing the cross section using MF1 and MF2 for the previously considered O 2 (v N2 (u

= 0) →

O2 (v

= 17)

+ N2 (u

= 1)

= 19) +

quasi-resonant transition and by plotting its values

against the value of the maximum impact parameter

bmax .

The cross section calculations

have been performed for two dierent sets of total (translational plus rotational) kinetic energy

U.

values of

U

For the rst set, whose outcomes are plotted in the lower panel of Fig. 2, the considered are 24.8, 37.2, and 62.0 meV. For the second set, whose outcomes are

plotted in the upper panel of Fig. 2, the values of

U

considered are 124, 248, and 496 meV.

The plotted values are those of the state to state cross section partially summed up to the given maximum impact parameter

bmax , σ(bmax ),

computed both on MF1 and MF2.

The plots clearly show that, apart from the dierent (of a few orders of magnitude) absolute values of the cross section for the two sets of total kinetic energy values, those computed on MF2 are signicantly larger than those computed on MF1 at low total kinetic energy (see the lower panel of Fig. 2) while the dierence decreases as the total kinetic energy increases (see the upper panel of Fig. 2).

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This eect is associated with the enhancement of the energy exchange due to the enhanced accessibility of the closer interaction region of MF2 (we shall further discuss this eect in Section 5 while performing a graphical analysis of the PESs). The plots show also that the percent contributions of high impact parameter collisions to the convergence value of the computed cross section is higher at lower energies while decreasing appreciably at higher energy. This conrms that quasi-resonant energy transfer mechanisms are to be largely attributed, as already illustrated in refs. 26,27,44,45, to larger b collisions and to the related orbiting capture eects determined by the protruding of the modied long range interactions into closer distances associated with the deformation of the collision partners. This is indeed an important conclusion of our investigation that rationalizes the occurrence of nonArrhenius eects at low temperature and clearly prompts a generalized use of QC techniques (instead of the QCT ones 27 ) at low collision energies and temperatures (depending also on the type of measurable quantities to be evaluated).

4 Further characterization of the non-Arrhenius behaviour In order to better understand the occurrence of the above illustrated non-Arrhenius behaviour we analyse here in more detail: a) the temperature dependence of the non reactive rate coecients, b) the interval of ΔE associated with quasi-resonant V-V energy transfer, and c) merits and demerits of adopting (simpler) QCT techniques with respect to the (more computationally demanding) QC ones in dealing with V-V and V-T energy transfers. As to item (a) the temperature dependence of quasi-resonant transitions rate coecients, we investigated in particular their low temperature variation for both vibrationally excited O2 and N2 at dierent ΔE values. In order to illustrate the outcomes of the QC calculations on MF2 we show in Table 2 the values of k(v = 10, u = 15, 25, 35|v = 11, u = u − 1). As to item (b) it is apparent from the various columns of Table 2 that the absolute value of the rate coecient increases as u increases in going from the left to the right hand side 12 ACS Paragon Plus Environment

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column of the Table. At the same time, the absolute value of the rate coecient decreases as

T lowers down from 800 K for all values of u with the absolute value and the slope depending on the value of u. However, it has to be noticed here that, while for columns u = 15 and 25 the decrease of the rate coecient as T lowers is continuous till T = 150 K, for u = 35 (whose ΔE is about −4.9 meV the dependence of the rate coecient on T has a change of sign around T = 600 K. Such non-Arrhenius inversion with temperature of the rate coecient has been already shown in Fig. 1 for the (19, 0|17, 1) transition (ΔE = 6.9 meV). The Table shows, therefore, that the quasi-resonant behaviour could persist up to an absolute value of

ΔE of the order of a couple of tens of meV. For completeness, the investigation has been extended also to the calculation of other state to state rate coecients for O 2 (v = 0) + N2 (u) collisions (typical of several dissociative plasmochemical systems in which O 2 (v) is deexcited by O atoms to the ground or low-lying vibrational states) similar to the V-V quasi-resonant transitions of the type shown in Fig. 2 though with larger ΔE . Table 3 shows the V-V rate coecients for N 2 deexcited to the next lower vibrational state and O 2 promoted from the ground vibrational state to the rst excited one. Here, again, for the transitions (0,25|1,24) and (0,30|1,29) ( ΔE = −10.7 and 7.1 meV, respectively) a non-Arrhenius inversion occurs around T = 500 K. Fig. 3 shows a subset of plots of the rate coecient dependence on temperature computed on the dierent PESs for the V-V (v = 13, 25, u = 0|v − 2, u = 1) and (v = 19, u = 0|v − 1, u = 1) transitions. In the lower panel of the gure the values of the rate coecients for the (19, 0|18, 1) transition computed on GB1, MF1 and MF2 denitely conrm the large dierence between GB1 and MF results (with an apparent similarity between the MF1 and the MF2 ones) over the whole temperature range together with their monotonous decrease as T lowers. The close similarity between MF1 and MF2 results is conrmed by the results shown in the central panel for the (25, 0|23, 1) transition. Such trend is further conrmed by the results plotted in the upper panel for the (13, 0|11, 1) transition. These results, however, whose ΔE is larger than 20 meV, show no inversion of slope at low temperature.

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Finally, for item (c) we checked also the possibility of replacing QC calculations by QCT ones in order to save time in massive simulations. However, it is well known that QCT calculations perform well at high energy and temperatures while they are quite inaccurate at low ones and in particular around threshold. For this reason we performed a parallel QC vs

QCT investigation at various temperatures. On this ground, we carried out a systematic comparison of the value of the rate coecients

computed on MF2 using both QC and QCT techniques and dierent initial states and temperatures starting from 19, 25, u = 0|v , u = 0)

T

equal to 1000 K. QC and QCT results obtained for the

(v =

V-T transitions in the mentioned temperature interval are given in

Table 4. These results conrm that, as found in ref. 19 and refs. 46 and 47 for N+N 2 and O+O2 reactions, QCT treatments can be satisfactorily used only at higher temperatures if tolerance criteria are softened and errors larger than one order of magnitude are accepted (as, however, is the case for several kinetic simulations). As a matter of fact, the results show that the set of values of the V-T rate coecient are in suciently good agreement in the 1000 − 5000 K range of temperature with the disagreement decreasing as

T

increases to

reduce to a factor of 2 at 5000 K. Completely dierent is the case of the whose QCT values computed at

(v = 19, 25, u = 0|v , u = 1)

T = 5000

V-V rate coecients

K are shown in Table 5. In fact, as apparent

from the Table, QCT rate coecients reasonably well reproduce the QC ones within one order of magnitude even if it has to be noticed that trajectory calculations do not show any excitation of N2 from the ground to the rst excited vibrational state at 3000 K. Indeed, at T

T = 1000

K and

= 1000 K the QCT rate coecients are smaller than the QC ones up to

about three orders of magnitude and suggest that some corrective actions need to be taken in order to have more accurate estimates. Further calculations have been performed with the purpose of more accurately testing the interval of condence when replacing QCT estimates of the rate coecients to the QC ones. For this purpose the multiquantum QCT and QC rate coecients calculated for the 14 ACS Paragon Plus Environment

Page 15 of 39

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(v = 10, u = 15, 25, 35|v , u ) processes at temperatures of 1000, 3000 and 5000 K are given

as supplementary material.

5 The progress variable representation of the interaction In this Section we trace back the evolution of our approach to progress variables for molecular processes and apply it to the title system. When a non reactive processes gets suciently close to the strong interaction region of the PES, excited and/or deformed congurations of the molecular system become involved. Let us consider rst the case of a slightly deformed molecule (say molecule b) and a strongly deformed one (say molecule a). When diatom a is suciently stretched it becomes more convenient to adopt a two particle-in-molecule (or a two pseudoatom ( pa)-bond pairs) formulation rather than the bond-bond one discussed in Section 2. This means that at intermediate values of the internuclear distance r of the molecular partners both VvdW and Velect must gradually evolve from a united- to a separated-atoms formulation of the interaction

as already done when considering the inclusion of the actual values of α(r) and Q(r) in MF2. Accordingly, in MF2 the VvdW and Velect interactions are partitioned into two pa1 -bond and pa2 -bond terms at all values of r (including r = re ) in order to ensure a smooth transition of

the system from diatom - diatom to excited diatom - diatom and then to dissociated diatom - diatom (or, correspondingly, from a relaxed "bond" to a stretched bond and then to a "near dissociated" molecular representation). This implies that the diatomic electric quadrupole Q(r) of N2 smoothly evolves as r increases into the corresponding asymptotic separated

atoms arrangements whose electronic conguration is symmetric and is associated with the ground state (4 S ) while that of O2 increases from a shorter distance negative value to about zero at r slightly larger than the equilibrium distance re and ends up, at dissociation, by generating two oxygen atoms in the non spherical 3 P symmetry. It has also to be pointed out here that, as the value of r gets comparable with that of R,

15



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

the formulation of

Velect

Page 16 of 39

of eq. 2 must be turned into a sum of Coulomb contributions dened

in terms of partial charges derived from a charge distribution on the molecular frame and

Q(r) is crucial for the denition of the dependence of the charge distribution

that the value of on

r.

Accordingly, the potential energy

V =

2

(i) Vintra (ri )

+

Rk−bond

can be formulated as:

2

i=1

in which

V

(k)

elect Vinter (Rk−bond , θk−bond ) + Vinter

(5)

k=1

is the distance between the pseudoatom

of the less deformed diatom and

θk−bond

pak

and the bond centre of mass

is the related in-plane angle. In eq. 5 the sum over

k

involves the two pseudoatom-bond pairs and represents the van der Waals component of the interaction that in our approach is usually formulated as an ILJ potential (please note that if

R = ∞, the dissociation energy

the energy zero is set as that of the two relaxed molecules at of both molecules needs to be added to

V ).

The electronic polarizability values obtained

in this way allow to estimate the potential parameters to be inserted in eq. 5 in order to formulate

(k)

Vinter

as an ILJ potential. In particular, the value of the ILJ parameters of the

two pseudoatom-bond pairs can be obtained by assigning to

αpa

the average value of

while the bond of the less deformed molecule use is described by and perpendicular ( ⊥) tensorial components

α (ri )

and

α⊥ (ri ),

α(ri )

α(ri )/2,

and its parallel ( )

respectively.

Transients between dierent channels can be associated to strongly deformed molecular congurations. In these cases, to the end of enforcing a proper behaviour on both short and long range shapes,

(i)

Vintra (ri ) (i)

has been formulated as:

(i)

(i)

Vintra (ri ) = VBO (ri )f (ri ) + VILJ (ri )(1 − f (ri )).

where

(6)

VBO (ri ) is a Bond Order polynomial 48 (usually truncated in our approach to the second (i)

order like Morse potentials) optimized to reproduce the close interaction regions. For the ILJ term of eq. 6, obtained from eq. 3 considering only the radial dependence, the following isotropic potential parameters: 31,49 for N2

ε

= 6.5 meV,

Rm

= 3.60 Å,

16


β

= 6.5 when using

Page 17 of 39

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the typical polarizability 3.44 Å,

β

= 1.1 Å3 of the nitrogen atom; 50 for the O2

α

= 6.5 when using the typical polarizability

modulation (switching) function

f (ri )

f (ri ) =

with

ri∗ = 5

Å and

di = 0.4

r

= 5.6 meV,

Rm

=

= 0.8 Å3 of the oxygen atom. 50 The

is given by the Fermi type expression:

1 1 + exp [(ri − ri∗ )/di ]

(7)

Å.

However, neither the intermolecular distance distances

α

ε

can be taken as a single appropriate

R

nor any of the involved internuclear

progress

variable to pivot the switching

between dierent channel oriented formulations of the interaction (not only between two dierent arrangements but also between long range and strong interaction regions). For this purpose one has to resort to a smooth and continuous adaptable combination of coordinates (like the polar formulation of the BO coordinates 51,52 ). In order to investigate the problem and dene a better diatom (Oa Ob

≡

progress variable for the title O 2 AB) pseudodiatom ([N] 2

in general to the (L-1)L + (L+1) Ob and L+1

≡

C

≡

→

≡

+ N2 processes let us refer to the generic

→

C) system AB + C

(L-1) + L(L+1) one (with L-1

≡

A + BC (or more

A

≡

Oa , L

≡

B

≡

[N2 ]) where for simplicity the N 2 molecule (frozen at its equilibrium

distance, aligned with one O atom of the O 2 molecule and coplanar with it) is considered as a pseudodiatom [N 2 ] and the angle

+ 1) ≡ Oa O ΦL ≡ (L − 1)L(L b [N2 ].

of the collinear ( ΦL = 180◦ ) encounter Oa Ob + [N2 ]

→

Isoenergetic contours

Oa + Ob [N2 ] on both MF1 (upper

panel) and MF2 (lower panel) are given in Fig. 4 as a function of the O a Ob distance, rOa −Ob , and the Ob [N2 ] distance,

rOb −[N]2 ,

(similar contours could, obviously, also be drawn using

related Jacobi coordinates). As can be seen from the gure, where the full reactive channel is represented, even if not involved in our study, the shape of the potential energy channel for a non reactive collision at xed

ΦO b

can be fairly simply described as a one dimensional diatomic potential function

(say a Morse like function) smoothly evolving as a function of the other diatomic internuclear

17



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Page 18 of 39

distance up to a certain extent into the close interaction region (and then evolving back). By comparing the MF1 and MF2 plots one can see that in the close distance region ( rOa −Ob ≈ reOa −Ob ≈ reOb −[N]2 ≈ rOb −[N]2 ) the two surfaces show dierences in singling out an easier

access to the strong interaction region on MF2 (of the order of 0.25 eV). However, when the system approaches the region of closer distances (or one wants to explicitly describe a closer interaction of one oxygen atom with the [N 2 ] pseudodiatom), individual diatomic internuclear distances are not a continuity variable suited to describe the evolution of the process even if reaction is excluded. For this reason one has to work out a more suitable progress variable smoothly switching between the two internuclear distances involved in the

representation even if limited to a short interval. When adopting the just mentioned internuclear distance representation, a popular way of approaching the problem is to adopt the so called "rotating model" in which at xed collision angle ΦL a one dimensional model potential (usually a Morse like one), initially coinciding with the entrance channel diatom (O a Ob ) and after a proper rotation coinciding with that of the exit channel pseudotriatom (O b [N2 ]), is used. The internuclear distance of such diatom is in its general formulation σL(r) dened as ∗ ∗ σL = [(r(L−1)L − r(L−1)L )2 + (rL(L+1) − rL(L+1) )2 ]1/2 (r)

(8)

∗ ∗ and centred on r(L−1)L ,rL(L+1) . The rotation angle ηL(r) is instead dened as

(r) ηL

= tan−1

∗ − rL(L+1) ) (rL(L+1)

∗ (r(L−1)L − r(L−1)L )

.

(9)

In eqs. 8 and 9 r(L−1)L and rL(L+1) are the already formulated two internuclear coordinates ∗ ∗ and r(L−1)L and rL(L+1) are the coordinates of the xed ΦL plane rotation centre (set to

have values larger (say a multiple) than the corresponding diatomic equilibrium distances in order to be placed on the classically forbidden region of the ridge separating the reactant from the product channel). Due to the modulus 3 property of the above used coordinates 18


Page 19 of 39

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the formalism applies to all the possible processes of the diatompseudodiatom. The piecewise nature of the above dened

progress

variable and the associated possible

articial loss of ux between the entrance and exit channels prompted the formulation of a better progress variable. A suitable alternative is represented by the already mentioned BO coordinates

ni

dened as

ni = exp [−βdiat (ri − rei )]

for two, three and four body sys-

tems. 13,48,5357 In BO representations, the physical unlimited space of the internuclear distances is mapped into the nite one of the related bond order coordinates in which the innity of the internuclear distance corresponds to the zero of the corresponding BO coordinate while the zero of the internuclear distance corresponds to a positive nite value of the corresponding BO coordinate. The related BO contour maps specic of the O 2 + N2 diatompseudodiatom analysis at xed collinear collision angle

ΦO b

are plotted in Fig. 5 by comparing side by side

the contour maps of MF1 and MF2. As apparent from the gure, a natural taken at dierent xed values of the angle

ΦL ( ΦO b

progress variable

for the title system) formed by the bond

of the reactant diatom (L-1)L of channel L and that of the chosen product one L(L+1), respectively (Oa

− Ob

and

Ob − [N2 ]

for the title system) is the smoothly and continuously

connecting the entrance and the exit channel polar angle as

n(L−1)L φL = arctan nL(L+1) instead of the previously dened angle distances). The coordinate associated to of the system) dened as:

(r)

ηL

φL

φ L ( φ Ob

for the title system) dened

(10)

(dened in terms of the related internuclear is the radius

ρL

1/2 ρL = n2(L−1)L + n2L(L+1)

(the related overall stretching

(11)

An obvious outcome of the BO representation is the formulation of the so called ROtating Bond Order (ROBO) model potential. 51 In the simplest cases the ROBO potential has been

19



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formulated as a low order polynomial in

ρL

and

φL

Page 20 of 39

as follows:

VLBO (ΦL ; φL , ρL ) = D(ΦL ; φL )P (ΦL , φL ; ρL )

in which

D(ΦL ; φL )

describes the evolution along

φL

of the xed

(MEP) of the potential energy channel from reactants (at At the same time, values of

ΦL

and

P (ΦL , φL ; ρL ) φL

ΦL

(12)

Minimum Energy Path

φL =0◦ ) to products (at φL =180◦ ).

describes the radial width of the potential channel at xed

that can be given simple polynomial formulations (like the harmonic

or proper anharmonic ones) respectful of the symmetry of the molecular system and easily switchable from a channel to a dierent one.

5860

An advantage of adopting the polar BO coordinates formalism (and in particular of the related

progress

variable

φL )

consists in the possibility of describing in a continuous way

the evolution of the system interaction from the entrance to the exit channel through the deformation of the colliding molecules into transients. This allows us to carry out a graphical analysis of the PESs inspect/compare the properties of the two MF PESs of the O 2 + N2 processes. The strong interaction region dierences of the two PESs are clearly evidenced by the contours.

They conrm, as already shown in Fig. 4 that the MF2 PES allows an

easier access of O 2 to the close interaction region and enhances so far the energy exchange mechanisms. Furthermore, the fact that the deformation can be univocally and continuously associated with the value of

φ Ob

in its continuous evolution between the two dierent asymptotic

channels allows us to dene along the collision angle MEP by varying relaxed

ρ

progress

variable

φ Ob

(for each value of

ΦO b )

the xed

until a minimum of the potential energy is reached.

61

The

ρ contour plots of the title system when considered as a diatom pseudodiatom (N 2

frozen and at equilibrium distance) are shown in the upper and lower panel of Fig. 6 for MF1 and MF2, respectively. The plots show that there are, indeed, dierences between the xed collision angle MEPs of the PESs that are better evidenced by the plot in the central

20


Page 21 of 39

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panel in which the dierence of the two graphs (MF2-MF1) is shown (the relaxation could be even pushed further by considering the N 2 orientation and/or internuclear distance). The shown dierences, however, conrm that MF2 allows a more close approach to the short distance regions of the interaction with a consequent more eective energy exchange oering a support to the rationalization of the extent of the non-Arrhenius eects on that PES.

6 Conclusions The detailed investigation reported in this paper on the processes of vibrational energy transfer in quasi-resonant collisions at thermal and subthermal regimes for the O 2 + N2 system has shown that related non Arrhenius behaviours are a typical quantum-like eect which need quantum-like (Quantum-Classical in our approach) to not go missed. As a result, we have been able to provide an accurate and comprehensive picture of the variation with temperature of the dependence of the quasi-resonant rate coecient on the vibrational energy gap between initial and nal vibrational states of the investigated system. Furthermore, a full characterization of the conditions where simpler QCT techniques, less computational demanding than QC ones, can be used is obtained. The investigation has also singled out the importance of using Open Molecular Science procedures in which data and competences of various laboratories cooperate in a completely clear and shareable fashion using a distributed molecular simulator allowing a detailed reproduction of the dierent steps of the procedure. In our case it has become apparent that an accurate formulation of the long range tail of the interaction needs to be coupled with a proper description of the closer range one in order to identify critical mechanisms like those leading non-Arrhenius behaviours. To this end we have also singled out the need for adopting suitable progress variables allowing a smooth and continuous description of the evolution of the molecular process. The benec impact of this strategy on both the construction of model potentials better describing the interaction in processes evolving from long to short range and a more proper representation

21



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Page 22 of 39

of the regions of the potential energy surface playing a signicant role in the determining energy transfer mechanisms even in absence of reactive processes has been also analysed for the O2 -N2 systems.

Acknowledgement F.P. acknowledges the Dipartimento di Chimica, Biologia e Biotecnologie dell'Università di Perugia and the "Fondazione Cassa Risparmio Perugia" (Codice Progetto: 2015.0331.021 Ricerca Scientica e Tecnologica). C.M. thanks The European Joint Doctorate on TCCM project ITN-EJD-642294-Theoretical Chemistry and Computational Modeling. E.G. acknowledges nancial support from the MINECO/FEDER of Spain under grant CTQ-201565033-P. Thanks are also due to the European Grid Infrastructure (EGI) for supporting the COMPCHEM Virtual Organization and the CMMST Virtual Research Community, and the OU Supercomputing Center for Education & Research (OSCER) at the University of Oklahoma (OU) for providing computing resources and services.

Supporting Information Vibrational state-to state quantum-classical and quasi-classical rate coecients for the

(v = 10, u = 15, 25, 35|v , u ) processes at temperatures of 1000, 3000 and 5000 K.

22


Page 23 of 39

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Hernández, M. I.;

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(44) Park, H.; Slanger, T. G. O 2 (X ,v

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= 8 − 22) 300 K quenching rate coecients for O 2

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Table 1: QC thermal rate coecients k(v=13, 19, u=0 | v=v − 2, u=1), in cm3 s−1, at T = 300 K computed on GB1, MF1, and MF2. Transition (13, 0|11, 1) (19, 0|17, 1)

Exp 44 GB1 MF1 MF2 2.2(-15) 5.9(-16) 4.5(-16) 1.2(-15) 1.4(-14) 5.1(-15) 2.5(-15) 1.3(-14)

Table 2: QC thermal rate coecients k(v=10, u=15, 25, 35 | v=11, u=u − 1), in cm3 s−1, computed on MF2. In brackets the energy gap ΔE between nal and initial vibrational state, in meV. T

/K

150 200 250 300 350 400 450 500 550 600 650 700 750 800

(10, 15|11, 14) (−75.9)

1.62(-16) 3.66(-16) 7.45(-16) 1.38(-15) 2.36(-15) 3.79(-15) 5.79(-15) 8.51(-15) 1.21(-14) 1.68(-14) 2.28(-14) 3.01(-14) 3.90(-14) 4.94(-14)

(10, 25|11, 24) (−40.4)

5.12(-14) 9.04(-14) 1.50(-13) 2.31(-13) 3.33(-13) 4.52(-13) 5.83(-13) 7.22(-13) 8.67(-13) 1.01(-12) 1.16(-12) 1.32(-12) 1.47(-12) 1.62(-12)

(10, 35|11, 34) (−4.9)


1.27(-11) 1.04(-11) 9.12(-12) 8.33(-12) 7.83(-12) 7.51(-12) 7.30(-12) 7.17(-12) 7.10(-12) 7.08(-12) 7.09(-12) 7.12(-12) 7.18(-12) 7.26(-12)


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 39

Table 3: QC thermal rate coecients k(v=0, u=5, 10, 15, 20, 25, 30 | v=1, u=u − 1), in cm3 s−1, computed on MF2. In brackets the energy gap ΔE between nal and initial vibrational state, in meV. T /K 150 200 300 500 800 1000 1500 2000 3000 5000

(0, 5|1, 4) (−81.7) 1.62(-18) 4.12(-18) 1.91(-17) 1.39(-16) 8.89(-16) 2.03(-15) 7.63(-15) 1.69(-14) 4.33(-14) 1.28(-13)

(0, 10|1, 9) (−64.0) 4.82(-17) 1.02(-16) 3.40(-16) 1.80(-15) 8.52(-15) 1.65(-14) 4.52(-14) 8.06(-14) 1.61(-13) 3.68(-13)

(0, 15|1, 14) (−46.2) 9.39(-16) 1.80(-15) 5.18(-15) 1.90(-14) 5.26(-14) 8.06(-14) 1.58(-14) 2.36(-13) 3.85(-13) 7.52(-13)

(0, 20|1, 19) (−28.4) 2.90(-14) 4.52(-14) 8.22(-14) 1.50(-13) 2.31(-13) 2.79(-13) 3.87(-13) 4.83(-13) 6.65(-13) 1.09(-12)

(0, 25|1, 24) (−10.7) 6.80(-13) 5.88(-13) 5.04(-13) 4.60(-13) 4.81(-13) 5.16(-13) 6.27(-13) 7.49(-13) 1.00(-12) 1.54(-12)

(0, 30|1, 29) (7.1) 5.82(-13) 5.40(-13) 4.99(-13) 4.86(-13) 5.31(-13) 5.80(-13) 7.20(-13) 8.58(-13) 1.13(-12) 1.65(-12)

Table 4: QCT and QC thermal rate coecients k(v=19, 25, u=0 | cm3 s−1, at T = 1000, 3000 and 5000 K computed on MF2.

v , u =0),

(19,0|16,0) (19,0|17,0) (19,0|18,0) (19,0|20,0) (19,0|21,0) (19,0|22,0)

T = 1000 K QCT QC 3.59(-16) 2.52(-16) 6.52(-15) 5.76(-15) 3.69(-13) 2.45(-13) 1.76(-13) 6.35(-14) 1.47(-15) 4.65(-16) 2.99(-17) 6.28(-18)

T = 3000 K QCT QC 1.43(-12) 5.00(-13) 4.99(-12) 2.52(-12) 2.67(-11) 1.61(-11) 2.01(-11) 1.08(-11) 2.97(-12) 1.18(-12) 7.61(-13) 1.61(-13)

T = 5000 K QCT QC 8.24(-12) 4.45(-12) 1.94(-11) 1.58(-11) 6.58(-11) 5.20(-11) 5.50(-11) 4.17(-11) 1.41(-11) 8.81(-12) 5.46(-12) 2.40(-12)

(25,0|22,0) (25,0|23,0) (25,0|24,0) (25,0|26,0) (25,0|27,0) (25,0|28,0)

1.03(-14) 1.17(-13) 2.29(-12) 1.21(-12) 3.01(-14) 1.80(-15)

4.83(-12) 1.26(-11) 4.79(-11) 3.76(-11) 8.00(-12) 2.69(-12)

1.54(-11) 3.13(-11) 8.83(-11) 7.55(-11) 2.32(-11) 1.02(-11)

Transition

4.65(-15) 5.45(-14) 1.07(-12) 3.34(-13) 6.20(-15) 1.80(-16)

2.26(-12) 7.98(-12) 3.30(-11) 2.33(-11) 4.02(-12) 8.75(-13)

32


1.20(-11) 3.11(-11) 8.88(-11) 6.57(-11) 1.93(-11) 7.19(-12)

in

Page 33 of 39

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


Table 5: QCT and QC thermal rate coecients k(v=19, 25, u=0 | cm3 s−1, at T = 5000 K computed on MF2. Transition (19,0|16,1) (19,0|17,1) (19,0|18,1) (19,0|19,1) (19,0|20,1) (19,0|21,1) (19,0|22,1)

QCT 1.07(-15) 1.27(-15) 2.21(-15) 2.61(-15) 1.61(-15) 1.27(-15) 5.35(-16)

Transition (25,0|22,1) (25,0|23,1) (25,0|24,1) (25,0|25,1) (25,0|26,1) (25,0|27,1) (25,0|28,1)

QC 1.76E-14 5.00E-14 3.04E-14 2.13E-14 1.17E-14 6.56E-15

QCT 1.27(-15) 1.54(-15) 2.08(-15) 2.14(-15) 1.81(-15) 6.70(-16) 4.69(-16)


v , u =1),

QC 2.63E-14 5.28E-14 2.44E-14 1.83E-14 1.19E-14 7.79E-15

in


O2(v =19) + N2(u =0) → Ο2(v’=17) + N2(u’=1) (ΔE=6.9 meV) 1e-13

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

Page 34 of 39

1e-14

1e-15 0

QC - GB1 QC - MF1 QC - MF2

1000

2000

3000

4000

5000

6000

T /K k(19, 0|17, 1) T


Page 35 of 39

10 MF2,U=496 meV

σ(bmax) /Å²

1

MF1,U=496 meV MF2,U=248 meV MF1,U=248 meV

0.1

MF2,U=124 meV

0.01 MF1,U=124 meV

0.001 -2

10

MF2,U=62.0 meV MF2,U=37.2 meV

σ(bmax

) /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


MF1,U=62.0 meV

-3

10

MF2,U=24.8 meV

MF1,U=37.2 meV

-4

MF1,U=24.8 meV

10

-5

10

1

2

3

4

5

6

bmax /Å

bmax 2v !2u"→2v # !2u $ % ΔE &' (

) ) * + , U bmax ' σ(bmax )

-. ACS Paragon Plus Environment


k /cm3 s-1

1e-13

O2(v =13) + N2(u =0) → Ο2(v’=11) + N2(u’=1) (ΔE=-28.7 meV)

1e-14 QC - MF1 QC - MF2 1e-15

1e-16 O2(v =25) + N2(u =0) → Ο2(v’=23) + N2(u’=1)

k /cm3 s-1

1e-13

(ΔE=42.6 meV)

1e-14 QC - MF1 QC - MF2 1e-15

1e-16 1e-13

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

Page 36 of 39

1e-15


1e-17 1e-19

O2(v =19) + N2(u =0) → Ο2(v’=18) + N2(u’=1) (ΔE=149 meV)

1e-21 0

1000

2000

3000

4000

5000

6000

T /K

k(13, 0|11, 1) k(25, 0|23, 1) k(19, 0|18, 1) T ACS Paragon Plus Environment

Page 37 of 39

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


Figure 4: Energy contours (in eV) of the MF1 (upper panel) and the MF2 (lower panel) PESs for the collinear process O a Ob + [N2 ] → Oa + Ob [N2 ] plotted as a function of the internuclear distances rO −O and rO −[N] (in Å). a

b

b

2



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 38 of 39

Figure 5: Energy contours (in eV) of the MF1 (upper panel) and the MF2 (lower panel) PESs for the collinear process O a Ob + [N2 ] → Oa + Ob [N2 ] plotted as a function of the BO coordinates nO −O and nO −[N] . a

b

b

2


Page 39 of 39

O2(v =19) + N2(u =0) → Ο2(v’=17) + N2(u’=1) (ΔE=6.9 meV) 1e-13

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


1e-14

1e-15 0


1000

2000

3000

4000

5000

6000

T /K


Closer versus Long Range Interaction Effects on ... - ACS Publications

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