Chemical Physics 351 (2008) 91–98

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Chemical Physics

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O–O2 state-to-state vibrational relaxation and dissociation rates based onquasiclassical calculations

Fabrizio Esposito a,*, Iole Armenise a, Giulia Capitta a, Mario Capitelli a,b

a IMIP-CNR (Istituto di Metodologie Inorganiche e dei Plasmi, Consiglio Nazionale delle Ricerche), Via Amendola 122/D, 70126 Bari, Italyb Chemistry Department, Bari University, Via Orabona 4, 70126 Bari, Italy

a r t i c l e i n f o

Article history:Received 17 January 2008Accepted 16 April 2008Available online 22 April 2008

Keywords:Vibrational energy exchangeRate coefficientsDissociationAtom–molecule collisionsOxygen

0301-0104/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.chemphys.2008.04.004

* Corresponding author. Fax: +39 080 592 9520.E-mail address: fabrizio.esposito@ba.imip.cnr.it (F.

a b s t r a c t

A complete set of V–T (vibration–translation) relaxation rates and of dissociation coefficients for the sys-tem O–O2 have been obtained by using quasiclassical trajectories on the Varandas and Pais potentialenergy surface. The results, averaged on a Boltzmann rotational distribution, cover the whole range ofthe vibrational ladder and are reproduced in closed form ready to be implemented in state-to-statekinetic models. The accuracy of the results has been tested by comparing them with available experimen-tal and theoretical values (ASI-CAST project is acknowledged).

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Non-equilibrium vibrational kinetics is a topic of increasinginterest due to its linking to different fields such as plasma-chem-istry, laser-chemistry, plasma-physics [1,2] and aerothermody-namics, the last matter including spacecraft reentry and relatedfields [3,4]. Sophisticated chemical physical models are beingdeveloped to control the complex phenomenology occurring inthe different fields [5,6]. In particular our group is developing astate-to-state approach, which consists in treating each vibrationallevel of the molecule as a new species with its own cross sections.This method requires complete sets of cross sections which in gen-eral are not available, even though a strong effort has been madeby different authors [7–9]. In particular we are performing massivecalculations of dynamical data involving the interaction of atomicspecies and molecules in vibro-rotational states with the aim ofcalculating cross sections and rates of vibrational relaxation anddissociation. To this end a quasiclassical trajectory method (QCT)has been developed [10] inserting in it available potential energysurface (PES) for important systems such as H2/H, N2/N, O2/O.The first two systems have been widely discussed by our group,leading to a systematic presentation of the relevant rates for theN2–N system [11]. The situation is scantier for the O2/O elementaryprocesses including vibrational deactivation by atomic oxygen (V–Ta) and dissociation process induced by atoms from each rovibra-tional state of electronic ground state of O2, i.e. the processes

ll rights reserved.

Esposito).

O2ðv; jÞ þ O! O2ðw; kÞ þ O ð1ÞO2ðv; jÞ þ O! 3O ð2Þ

where v and w are, respectively, initial and final vibrational states,while j and k are initial and final rotational states. The rates of pro-cess (1) have been reported by Lagana’ et al. in [12] by means ofquasiclassical method (QCT) on the same PES used in this paper,limiting however, the range of the vibrational quantum numberfrom v = 5 to v = 30. Dissociation was completely neglected. Theserates, even though very important for non-equilibrium vibrationalkinetics, are not sufficient for a complete characterization of theO2/O system from the point of view of the state-to-state kinetics.Therefore we have coped with a series of extensive calculationsto construct a database of cross sections for both the processes(1) and (2), considering each rotational state individually (not ina rotational distribution) and the explicit dependence on transla-tional energy (not a maxwellian distribution, as in rate calcula-tion). Preliminary results have been reported for the dissociationprocess in [13], while more recently we have discussed our QCTV–Ta rates by comparing them with available experimental data[14].

In this paper, we present interpolation formulae for our QCTrate coefficients of the two processes as a function of rotransla-tional temperature and initial and final vibrational quantum num-bers. These formulae are of particular importance for the inclusionof rate coefficients in fluidynamic codes. V–Ta rates will be vali-dated with the existing Lagana’s rates [12], while the dissociationrates will be validated against experimental thermal dissociationrates.

Table 1Vibrational levels (for j = 0) in eV used in this work

Vibrational quantum number Energy (eV)

0 �5.11531 �4.92212 �4.73153 �4.54334 �4.35785 �4.17496 �3.99487 �3.81758 �3.64319 �3.4717

10 �3.303411 �3.138112 �2.976013 �2.817214 �2.661715 �2.509616 �2.361017 �2.216018 �2.074619 �1.936920 �1.803021 �1.672922 �1.546923 �1.424924 �1.307125 �1.193526 �1.084227 �0.9793928 �0.8791329 �0.7835430 �0.6927431 �0.6068532 �0.5260133 �0.4503634 �0.3800435 �0.3152236 �0.2560737 �0.2027738 �0.1555039 �0.1144540 �0.07981841 �0.05175142 �0.03032343 �0.01539744 �0.006374845 �0.001926146 �0.00029275

O2 potential well depth in eV: 5.21275 (energy has the zero value at dissociation).

92 F. Esposito et al. / Chemical Physics 351 (2008) 91–98

2. Molecular dynamic calculations

We started quasiclassical calculations of O + O2 collision pro-cesses since 2002 [13] with specific interest for dissociation, thenwe largely extended such calculations in [14], showing results con-cerning V–Ta rate coefficients. In both cases we found good agree-ment with experimental results, which is important for assessingthe validity of both the potential energy surface (PES) and thedynamical method used, and also encouraging for extending fur-ther calculations, especially enlarging the range of kinetic energyconsidered. The quasiclassical calculations have been performedutilizing a code developed in our group and improved in the lastten years by adding parallel capabilities and the possibility of usingthis software on computational grids. In the case of oxygen, due tothe quite complex potential energy surface (the DMBE PES ofVarandas and Pais [15]), it is extremely important to deeply exploitthe possibilities offered by large computational grids [16], in whichspecial complementary software must be developed in order todistribute computational loads by appropriately segmenting thecalculations, but also to check that each segment has been cor-rectly executed (the probability of failure in a large grid environ-ment can be 10–20% or more), eventually recalculate it onfailure, and finally retrieve and correctly add the outcome to thelarge mass of the other results. Quasiclassical method fits perfectlyto the distributed computational model, because of its intrinsicseparability in small amounts of calculations. On the used PESwe have calculated the whole ladder of rovibrational states byWKB method, obtaining 47 vibrational levels (Table 1) and a max-imum number of rotational levels (reached for v = 0) of 237 (the to-tal number of rovibrational states amounts to about 6400). It is ofrelevance the accurate calculation of energy levels, as pointed outin Lino da Silva et al. [17].

We used a continuous range of translational energy from 10�3

to 3 eV, a density of 2000 trajectories per Å of impact parameterand per eV of kinetic energy, with application of stratified sam-pling. These values are by far lower than those used for past calcu-lations on hydrogen and nitrogen [18], but in the oxygen case thelimiting factor is the long computational time required for eachtrajectory in comparison to the other species, due to the heavy loadintroduced by the PES calculation. We performed an error analysison rate coefficients, starting from statistical error on cross sectiondata. For V–Ta processes errors are generally around or better than10%, while for dissociation error value is variable along the temper-ature axis, with an ascending trend similar to that of the corre-sponding rate but less steep, with a crossing at a rate value nothigher (often quite less) than 10�12 cm3/s. This means that dissoci-ation rate values higher than 10�12 cm3/s are surely above the cor-responding error value. Given the interval of kinetic energyconsidered, the range of temperature for rate coefficients shouldbe 300–5000 K, even if an extension up to 10000 K can be per-formed without large errors for V–Ta rates, due to the relativelyconstant rate value as a function of temperature. On the contrarydissociation rates are generally strongly increasing with tempera-ture, therefore an extension should be considered with a certaincaution. The reverse is valid for temperature values lower than300 K: in this case dissociation is negligible, while V–T rates canbe important, especially for initial low lying vibrational states.Moreover, at very low energy, classical dynamics is in any casescarcely reliable.

Starting from v = 0, one vibrational state in five has been consid-ered, plus some other intermediate states of particular interest fora good interpolation: v = 1–3–5–7 and each state from 39 to 46.The cross sections are obtained by discretizing the energy axis into1000 subintervals in order to have sufficiently good determina-tions of thresholds. A degeneracy factor of 1/27 is applied to V–

Ta rate coefficients, as suggested in [15]. This is explicitly evi-denced in the fitting formulae of the following paragraph. ThisV–Ta degeneracy factor has been formulated on the basis of Truh-lar discussion in [19] by the authors of the PES used in this work. Itis linked to the degeneracy of the reactants and the PES and tothe hypothesis of adiabaticity for the studied reaction. Its valueis still an open problem in the scientific community, for exam-ple Billing [20] suggests a temperature dependent factor: 1/(3�(5 + 3�exp(�227.6/T) + exp(�325.9/T))), which asymptoticallyreaches the 1/27 value. For dissociation a variable degeneracy fac-tor is applied, as explained in [13], in order to take into accountapproximately the excited electronic states of O2. Both thesedegeneracy factors are independent of either the PES or thedynamical method used, and within the range of validity of therespective approximations they are exact numbers, not adjustableparameters.

Consideration of all possible rovibrational states as final statesis impossible from a practical point of view, because of the unac-ceptable number of trajectories required, but also for the enormouskinetic code which should be set up for using those data. Instead

Table 2Coefficient c1jk for the calculation of a1 (v, Dv)

j,k Dv = 1 Dv = 2–10 Dv = 11–20 Dv = 21–30

1,1 �26.18993227 �19.45675583 �3.993663 2.8217591,2 0 �3.38007590 2.684515 4.0831381,3 0 89.85158436 �1.009927e5 �8.809991e11,4 0 5.853646e�2 �0.283676 2.369644e�22,1 �1.69828917 �1.48712025 �3.030965 3.8410802,2 0 �0.54018334 �4.594443 �0.9986372,3 0 �53.33457116 3.590903e4 �3.438479e12,4 0 �9.543789e�2 7.104764e�2 1.222381e�23,1 3.349076e19 1.505136e21 5.492061e21 4.330479e213,2 0 1.621622e21 1.212196e21 �1.677646e223,3 0 �1.066183e21 9.092488e21 �5.573334e213,4 0 2.169160e20 1.540038e20 �3.089812e19

F. Esposito et al. / Chemical Physics 351 (2008) 91–98 93

we obtained cross sections from each initial rovibrational state to-wards each final vibrational state w, summing contributions fromany final rotational state k compatible with w.

An example of rotationally averaged monoquantum deactiva-tion cross sections is given in Fig. 1a–b. Actually we have smoothedcross sections with a local cubic spline smoothing procedure,which is necessary for the very thin channels of translational en-ergy discretization. Unfortunately it is impossible to follow exactlythe extremely fast and large variation of some V–Ta cross sectionson the energy axis, so very high values are represented withapproximation. A specific study is in progress to cope with thisimportant problem. The aim is that of solving both the problemsof high accuracy and low storage space of cross sections, whichis of relevance also in the application in numerical codes like a di-rect Monte Carlo, because of the enormous mass of data to manage.In Fig. 1a we have set the initial vibrational state to v = 1, with twopossible values of rotational temperature: Trot = 300 K, 10,000 K, inorder to show its effect on cross section. This means that many ini-tial rotational states are involved in the calculation, particularly athigh temperature. It is clear that increasing Trot causes an impor-tant decreasing (up to a factor 3 for very low energy values) ofcross section for translational energy less than 1.5 eV, while forhigher energies the effect is negligible. In Fig. 1b the comparisonat a fixed Trot = 300 K is among different initial vibrational states:v = 1, v = 15, v = 30. It is clear the higher values of the v = 1 crosssection with respect to the other two, while the difference is verylimited between v = 15 and v = 30 cross sections.

From cross section data we then calculated rate coefficientsfrom a given vibrational state to each possible final vibrationalstate, averaging initial rotation on a given rotational temperature.In all the calculations presented here, rotational temperature isequal to translational temperature.

4,1 �3.946126e20 �1.532916e20 1.308503e20 �1.194045e204,2 0 �4.105380e19 1.831856e20 �5.121704e194,3 0 1.185042e21 1.079540e22 4.013656e214,4 0 �1.748646e18 �5.608629e18 �1.052730e185,1 1.391056e19 �4.838473e18 2.160753e19 �9.939380e185,2 0 9.529399e18 �1.465914e19 4.180038e185,3 0 5.290114e19 5.483520e21 �2.265448e185,4 0 6.021495e16 1.142128e17 1.050411e16

The first column reports the indices j,k of each coefficient c1jk, then each successivecolumn label indicates the range of the corresponding c1jk applicability.

3. Interpolated vibrational–translational rates

The rate coefficients of the V–Ta process

O2ðvÞ þ O! O2ðwÞ þ O; with v > w

have been successfully interpolated by means of the followingformula:

0

10

20

30

40

50

0 1 2 3

Trot

= 300 K

Trot

= 10000 K

cros

sse

ctio

n(Å

2)

energy (eV)

v = 1a b

Fig. 1. Rotationally averaged monoquantum deactivation cross sections versus the translinitial v = 1, 15, 30 (b).

KVTðT; v;DvÞ ¼ DegF � expða1ðv;DvÞ þ a2ðv;DvÞ= lnðTÞþ a3ðv;DvÞ � lnðTÞÞ; ð3:aÞ

where Dv is (v � w) and where

aiðv;DvÞ ¼ bi1ðDvÞ þ bi2ðDvÞ � lnðvÞ þ ðbi3ðDvÞ þ bi4ðDvÞ � v

þ bi5ðDvÞ � v2Þ=ð1021 þ expðvÞÞ; ð3:bÞ

bijðDvÞ ¼ cij1 þ cij2 � lnðDvÞ þ cij3 � Dv � expð�DvÞþ cij4 � Dv � Dv: ð3:cÞ

The coefficients cijk have been generated using a linear least squaresmethod and reported on Tables 2–4 (respectively, c1jk, c2jk, c3jk).

DegF is the degeneracy factor for V–Ta transitions (see Section2).

The temperature (T = 50–10,000 K) and initial vibrational state(v = 1–46) ranges considered are the same as those of the original

0

10

20

30

40

50

0 1 2 3

v = 1

v = 15

v = 30

cros

sse

ctio

n(Å

2)

energy (eV)

Trot

= 300 K

ational energy from v = 1 at Trot = 300 K and 10,000 K (a), and holding Trot = 300 K for

Table 3Coefficient c2jk for the calculation of a2 (v, Dv)

j,k Dv = 1 Dv = 2-10 Dv = 11-20 Dv = 21-30

1,1 7.83331061 14.47821980 �7.575157e1 �1.066105e21,2 0 �43.32225364 �9.234026 6.618737e11,3 0 �348.16803354 2.807916e5 2.630406e21,4 0 �0.16418596 0.333397 2.791153e�22,1 3.71221451 32.66821082 �7.713850 �2.658559e12,2 0 �0.18462189 2.545634e1 �9.1672112,3 0 119.03129957 �9.592245e4 1.678357e22,4 0 0.22258852 �0.179262 �0.1064743,1 3.573261e20 1.522507e21 4.002520e21 1.312884e223,2 0 2.654567e21 �8.192010e21 �5.437653e213,3 0 �3.528669e21 1.462011e23 5.735816e213,4 0 �2.861293e20 �1.821224e20 1.568233e204,1 6.433503e20 �1.533872e21 �2.912948e21 4.530951e214,2 0 �1.522587e20 6.399791e19 2.662341e204,3 0 �3.124046e21 �3.531505e22 �2.932068e224,4 0 2.322209e19 1.964744e19 6.788371e185,1 �2.901352e19 �3.762650e19 �7.070723e19 �3.473472e195,2 0 1.955942e19 4.805948e19 �5.623449e185,3 0 8.847719e18 5.201014e21 2.765213e205,4 0 �8.252347e17 �4.170396e17 �6.030509e16

The first column reports the indices j,k of each coefficient c2jk, then each successivecolumn label indicates the range of the corresponding c2jk applicability.

Table 4Coefficient c3jk for the calculation of a3 (v, Dv)

j,k Dv = 1 Dv = 2–10 Dv = 11–20 Dv = 21–30

1,1 0.37163948 0.88657647 �1.271181 �3.4768251,2 0 �1.14258799 0.317834 �0.4156751,3 0 �4.53080420 5.830886e3 2.341590e11,4 0 4.732032e�3 7.186328e�3 �1.760866e�32,1 0.10587091 �0.18566919 �0.709028 �0.5624112,2 0 �0.19256423 4.753706e�2 0.1663192,3 0 3.95819117 �1.757570e3 2.3566592,4 0 1.353007e�2 1.465161e�3 �7.409818e�43,1 �5.312491e19 �2.027215e21 �8.686388e20 1.908092e213,2 0 2.381051e21 9.428176e20 1.107010e213,3 0 �1.248596e20 1.738719e22 �1.769244e223,4 0 �4.014395e19 �1.251868e19 1.272578e184,1 3.754092e19 1.819921e19 1.877581e19 �2.139241e194,2 0 3.708631e19 1.097908e19 2.171483e194,3 0 2.031805e19 �1.006633e22 �2.478535e194,4 0 �6.021031e17 2.734279e16 7.612186e165,1 �1.189832e18 1.530423e18 2.238756e18 1.542813e185,2 0 �1.859900e18 �7.688157e17 �7.531694e175,3 0 �8.207403e18 �2.061752e21 �4.924709e185,4 0 9.708459e15 �3.427940e15 �5.546602e14

The first column reports the indices j,k of each coefficient c3jk, then each successivecolumn label indicates the range of the corresponding c3jk applicability.

94 F. Esposito et al. / Chemical Physics 351 (2008) 91–98

QCT rates. On the other hand, due to the huge spread in the rates,we made use of not a unique set of interpolation coefficients cijk forthe vibrational quantum jump Dv that runs from 1 to 30. Actually

Table 5Maximum error of the interpolated QCT V–Ta rates at different translational temperature

Tmin (K) Dv = 1 2 6 Dv 6 10 11 6 Dv 6 20

50 0.22576(T = 150, v = 10)

1.17791(T = 200, v = 3, Dv = 3)

1.11371(T = 50, v = 4

300 0.16988(T = 300,v = 10)

0.23405(T = 300, v = 46, Dv = 2)

0.43323(T = 300, v =

500 0.10491(T = 500, v = 3)

0.21422(T = 500, v = 46, Dv = 2)

0.38245(T = 500, v =

1000 9.61804e�2(T = 1000, v = 25)

0.20707(T = 1000, v = 46, Dv = 2)

0.35429(T = 1000,v =

The (T, v, Dv) values at which the maximum has been found are reported in parenthesis

there are four different sets of coefficients: one for Dv = 1 (secondcolumn on Tables 2–4), one for Dv = 2–10 (third column), one forDv = 11–20 (fourth column), one for Dv = 21–30 (fifth column).

The adopted interpolation formula well reproduces the QCT V–Ta rates, as it is confirmed by the error analysis. The error in the fit-ting of each rate has been obtained as the absolute value of the dif-ference between the logarithms of the QCT rate (RVT (T, v, Dv)) andthe fitted one (KVT (T, v, Dv)), as follows:

errðT; v;DvÞ ¼ jlog10ðRVTðT; v;DvÞÞ � log10ðK

VTðT; v;DvÞÞj:

At this stage the calculation of the maximum error has been consid-ered as the maximum among the errors of the considered rates. Onthe other side, the root mean square of fit error has been calculatedas

RMS ¼ sqrtX

T;v;DvðerrðT; v;DvÞ2Þ=n

� �ð4Þ

with n number of terms in the sum, i.e. number of rates included inthe error evaluation.

In particular the maximum error (Table 5) and the RMS (Table6) have been evaluated subdividing the Dv range into smaller sets:Dv = 1, 2 6 Dv 6 10, 11 6 Dv 6 20, 21 6 Dv 6 30,1 6 Dv 6 30, as re-ported in columns 2–6 of Tables 5 and 6. It is important to stressthat the range 1 6 Dv 6 30 contains four different sets of coeffi-cients cijk. Four different translational temperature ranges havebeen chosen, fixing the minimum T value at 50, 300, 500, 1000 K,while the maximum value in each interval is fixed to 10,000 K.Moreover on Table 5, under each maximum error value, the corre-sponding T, v and Dv values have been reported in parenthesis.Inspection of Table 5 shows that at low temperature the errorscan be quite large. Similar considerations apply to the very largevibrational quantum jump transitions. These considerations aregenerally confirmed by the RMS analysis (Table 6). In particularthe maximum RMS is approximately 50% of the rate value(RMS = 0.174, in the range: T P 50 K, 21 6 Dv 6 30), whereas, theminimum RMS is 7% of the rate value (RMS = 0.03, in the range:T P 1000 K, Dv = 1). In any case in the whole range 1 6 Dv 6 30the RMS at T P 1000 K is 17% of the rate value (RMS = 0.069): webelieve that this is a satisfactory result, given the statistical errorson the original rates (see Section 2).

A sample of QCT rates and corresponding interpolated V–Tarates have been reported in Figs. 2–5. The rate coefficients, re-ported versus T, correspond to initial vibrational levels v (1, 10,20, 30, 40, 46) and vibrational quantum jumps Dv (Dv = 1 inFig. 2, Dv = 10 in Fig. 3, Dv = 20 in Fig. 4, Dv = 30 in Fig. 5). In generalwe can observe a satisfactory agreement between QCT and inter-polated results.

Fig. 6 reports a comparison of our interpolated rates with thecorresponding data calculated by Lagana’ [12] for monoquantumtransitions. Once again the agreement is very satisfactory.

s ranges and vibrational quantum jump range

21 6 Dv 6 30 1 6 Dv 6 30

5, Dv = 15)3.35680(T = 50, v = 46, Dv = 22)

3.35680(T = 50, v = 46,Dv = 22)

46, Dv = 18)0. 7056(T = 300, v = 46, Dv = 23)

0.7056(T = 300, v = 46, Dv = 23)

46, Dv = 13)0. 63523(T = 500, v = 46, Dv = 23)

0.63523(T = 500, v = 46, Dv = 23)

46,Dv = 13)0. 46588(T = 1000,v = 46,Dv = 23)

0.46588(T = 1000,v = 46,Dv = 23)

.

Table 6Root mean square of the logarithm of the V–Ta rates fit error at different translationaltemperatures ranges and vibrational quantum jump ranges

Tmin(K) Dv = 1 2 6 Dv 6 10 11 6 Dv 6 20 21 6 Dv 6 30 1 6 Dv 6 30

50 6.245e�2 0.122 0.108 0.174 0.132300 4.479e�2 6.522e�2 8.590e�2 9.696e�2 8.1e�2500 3.790e�2 6.316e�2 8.333e�2 8.271e�2 7.485e�21000 3.009e�2 6.021e�2 8.186e�2 6.686e�2 6.873e�2

F. Esposito et al. / Chemical Physics 351 (2008) 91–98 95

4. Interpolated dissociation rates

The dissociation rate coefficients of the reaction Eq. (2) can besatisfactorily reproduced as a function of the translational temper-ature and of the initial vibrational level v by the formula

Kd-stsðT; vÞ ¼ expða1ðvÞ þ a2ðvÞ=T þ a3ðvÞ � lnðTÞÞ; ð5aÞ

10-13

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito; v = 1

Esposito; v = 10

Esposito; v = 20

Interp.; v = 1

Interp.; v = 10

Interp.; v = 20

k v->

w(c

m3s-1

)

T

(v-w) = 1a b

Fig. 2. V–Ta rate coefficients by QCT calculations (curves) and by interpolation formulasjump Dv = 1.

10-13

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito; v = 10

Esposito; v = 20

Interp.; v = 10

Interp.; v = 20

k v->

w(c

m3s-1

)

T

(v-w) = 10

a

a

Fig. 3. Same as in Fig. 1 for a vibra

where

aiðvÞ ¼ bi0 þ bi1 � vþ bi2 � v2 þ bi3 � v3 þ bi4 � v4 þ bi5 � v5

þ bi6 � v6 þ bi7 � v7: ð5bÞ

The relevant bij coefficients are reported in Table 7. The interpola-tion has been performed in the temperature range T = 50–10,000 K describing the complete set of vibrational levels v = 0–46. The dissociation coefficients are significantly smoother thanthe V–Ta coefficients and their trend as a function of temperatureis very simple. As a consequence, the fit of the dissociation ratecoefficients gives better results than the fit of the V–Ta rates, asdemonstrated by the error analysis.

As well as in the V–Ta case, the error in the calculation of eachdissociation rate has been calculated as the absolute value of thedifference between the logarithms of the QCT rate (Rd-sts(v,T))and of the fitted one (Kd-sts(v,T))

10-13

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito.; v = 30Esposito.; v = 40Esposito.; v = 46

Interp.; v = 30Interp.; v = 40Interp.; v = 46

k v->

w(c

m3s-1

)

T

(v-w) = 1

(points) versus T at different initial vibrational level v and for a vibrational quantum

10-14

10-13

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito; v = 30

Esposito; v = 40

Esposito; v = 46

Interp.; v = 30

Interp.; v = 40

Interp.; v = 46

k v->

w(c

m3s-1

)

T

(v-w) = 10

b

b

tional quantum jump Dv = 10.

10-13

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito; v = 20

Interp.; v = 20

k v->

w(c

m3s-1

)

T

(v-w) = 20

a

10-15

10-14

10-13

10-12

0 2000 4000 6000 8000 10000

Esposito; v = 30

Esposito; v = 40

Esposito; v = 46

Interp.; v = 30

Interp.; v = 40

Interp.; v = 46

k v->

w(c

m3s-1

)

T

(v-w) = 20

b

a b

Fig. 4. Same as in Fig. 1 for a vibrational quantum jump Dv = 20.

10-15

10-14

10-13

0 2000 4000 6000 8000 10000

Esposito; v = 30

Esposito; v = 40

Esposito; v = 46

Interp.; v = 30

Interp.; v = 40

Interp.; v = 46

k v->

w(c

m3s-1

)

T

(v-w) = 30

Fig. 5. Same as in Fig. 1 for a vibrational quantum jump Dv = 30.

10-13

10-12

10-11

0 10 20 30 40 50

T=1000 K; Esposito

T=4000 K; Esposito

T=1000 K; Lagana'

T=4000 K; Lagana'

kvTa v

->v-

1(c

m3s-1

)

v

T = 4000

T = 1000

a b

Fig. 6. Lagana’ et al. and interpolated Esposito et al. monoquantum V–Ta rates either vvibrational levels (b).

Table 7bij Coefficients for the calculation of ai (v)

j/i 1 2 3

0 �27.0145 �59620 0.6754551 �3.19019 2178.93 0.3766822 1.05845 �11.2525 �0.1215983 �0.126656 1.59843 0.01448644 0.00721311 �0.177455 �0.000823525 �0.000211765 0.00687121 2.41602e�056 3.10206e�06 �0.000119773 �3.53967e�077 �1.79668e�08 7.84053e�07 2.05237e�09

The labels on the first column are referred to the indices j of each coefficient; thelabels on the first row are referred to the indices i of each coefficient, then at thesecond, third, fourth columns correspond a1(v), a2(v), a3(v), respectively.

96 F. Esposito et al. / Chemical Physics 351 (2008) 91–98

errðT; vÞ ¼ jlog10ðRd-stsðv; TÞÞ � log10ðKd-stsðv; TÞÞj:

Finally the maximum error and the root mean square have been cal-culated; this last reads

RMS ¼ sqrtX

T;vðerrðT; vÞ2Þ=n

� �; ð6Þ

where n is the number of rates included in the error evaluation.

10-12

10-11

0 2000 4000 6000 8000 10000

Esposito; v = 5

Esposito; v = 30

Lagana'; v=5

Lagana'; v=30

kvTa v

->v-

1(c

m3s-1

)

T

ersus the vibrational energy at T = 1000, 5000 K (a) or versus T at different initial

Table 8Maximum error and root mean square of the dissociation rates (cm3 s�1) fit atdifferent translational temperatures ranges

Tmin(K) Maximum error T of maximum error v of maximum error RMS

50 4.1772 90 1 0.3723300 0.5795 300 5 0.0878500 0.3707 500 5 0.05851000 0.1555 10,000 20 0.0488

Table 9Coefficients of the Arrhenius-like dissociation rate Kd(T) (in cm3 s�1)

A B h

Present work 6.1953e�6 �0.55724 60,496Park 1.66044e�2 �1.5 59,500Shatalov 1.393109e�5 �0.645 59,874

Table 10Examples of dissociation rates (cm3 s�1), at some fixed T, calculated by differentauthors using the Arrhenius-like formula (8) or present work total dissociation rateKd(T) calculated using formula (8)

T Park Shatalov Present work

1000 7.5800e�33 1.6100e�33 7.0350e�342000 2.2300e�20 1.0300e�20 6.5472e�213000 2.4600e�16 1.7100e�16 1.2496e�164000 2.2800e�14 2.0900e�14 1.6466e�145000 3.1900e�13 3.6100e�13 2.9939e�136000 1.7600e�12 2.3600e�12 2.0318e�127000 5.7700e�12 8.9000e�12 7.8200e�128000 1.3700e�11 2.3800e�11 2.1600e�119000 2.6200e�11 5.0600e�11 4.6700e�1110,000 4.3300e�11 9.2000e�11 8.6300e�11

F. Esposito et al. / Chemical Physics 351 (2008) 91–98 97

In Table 8 the maximum error (column 2), its corresponding Tand initial vibrational level v (columns 3, 4) and the RMS (column5) have been reported considering different temperature windows(the lower temperature value is reported in the first column, thehigher temperature value is T = 10,000 K for each window).

All dissociation rates have been included in the maximum errorand in the RMS evaluation reported in Table 8, regardless of theirmagnitude.

The RMS error ranges between a factor 2.36 with respect to therate value and 12% of the rate value, the better result being relativeto the 1000–10,000 K temperature range, whereas the worse oneis relative to the whole range 50–10,000 K. It seems clear thatthe interpolation formula is worse at lower temperatures, wherethe rates present very small values. As a consequence, neglectingthe dissociation rates lower than 10�30 cm3 s�1 leads to thedecrease of the RMS and maximum errors: the RMS error is about12–14% of the rate value, and the result relative to the differenttemperature ranges are more or less equal to each other in this case.

In Fig. 7 the QCT dissociation rates and their correspondinginterpolated rates, are plotted for different initial vibrational levels(0, 10, 20, 30, 40, 44, 45, 46) versus temperature. The interpolationformula fits the data obtained by QCT calculations in a satisfactoryway.

To validate the present state-to-state dissociation rates it is pos-sible to average interpolation of QCT rates with a Boltzmann vibra-tional distribution and compare the result with existingphenomenological global rates [21]. In order to perform this com-parison, the following equation has been used:

KdðTÞ ¼X

vðO2ðvÞ=O2ÞKd-stsðv; TÞ ð7Þ

where O2 is the total molecular population density, and O2(v) is thepopulation density of the v-th vibrational level obtained by allow-ing a Boltzmann distribution. The corresponding results can be eas-ily fitted by an Arrhenius expression

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

1000 10000

Esposito; v=0Esposito; v=10Esposito; v=20Esposito; v=30Interp.; v=0Interp.; v=10Interp.; v=20Interp.; v=30

k dO

2(v)

+O

->3O

(cm

3s-1

)

T

a

Fig. 7. Dissociation rate coefficients by QCT calculations (curves) and by inter

KdðTÞ ¼ ATB expð�h=TÞ: ð8Þ

The relevant A, B and h coefficients calculated in the present workhave been reported in Table 9 and compared with the semiempiri-cal values of Shatalov [22] and Park [6] as reported by [21].

Dissociation rates calculated by Eq. (8) considering the differentparameters of Table 9 have been reported in Table 10. Inspection ofthis table shows a satisfactory agreement between our global ratesand the corresponding values obtained by Park and Shatalov thusindicating a fair amount of confidence in our state-to-state dissoci-ation rates.

5. Conclusions

Cross sections and rates for vibrational relaxation and dissocia-tion processes of vibrotational O2 molecules colliding with oxygenatoms have been calculated by running QCT method on an appro-priate PES. The results have been reported in analytical forms eas-ily implementable in state-to-state models considering non-equilibrium vibrational kinetics of oxygen and air molecules. The

10-11

10-10

10-9

10-8

10-7

1000 10000

Esposito; v=40Esposito; v=44Esposito; v=45Esposito; v=46

Interp.; v=40Interp.; v=44Interp.; v=45Interp.; v=46

k dO

2(v)

+O

->3O

(cm

3s-1

)

T

b

polation formulas (points) versus T at different initial vibrational level v.

98 F. Esposito et al. / Chemical Physics 351 (2008) 91–98

accuracy of the results has been validated against available exper-imental and theoretical values.

The present data on O–O2 system, together with recent databased on N–N2, provide complete sets of rates to be used in thenon-equilbrium vibrational kinetics of air plasmas filling a gap inthis description due to the practical absence of these rates. Inclu-sion of such rates in the kinetic codes should strongly modify theform of O2 and N2 vibrational distributions under non-equilibriumconditions.

Acknowledgements

The present paper has been partially supported by ASI underCAST project and by MIUR FIRB (Project No. RBAU01H8FW_003).

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