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Chemical Physics Letters 443 (2007) 222–226

The relaxation of vibrationally excited O2 molecules by atomic oxygen

Fabrizio Esposito a,*, Mario Capitelli a,b

a Institute of Inorganic Methodologies and Plasmas of C.N.R. – via Amendola 122/D, 70126 Bari, Italyb Department of Chemistry, University of Bari, via Orabona 4, 70126 Bari, Italy

Received 12 February 2007; in final form 18 June 2007Available online 28 June 2007

Abstract

A complete set of V–T (vibration–translation) relaxation rates of vibrationally excited oxygen molecules by parent atoms has beenobtained by running quasiclassical trajectories on an accurate potential energy surface. The results have been validated by comparingthem with old and new experimental data. Satisfactory agreement has been obtained with recent low temperature experimental data.Comparison with older rate coefficients obtained at higher temperature prompts the elaboration of new experiments not based on theapplication of models restricted to monoquantum transitions. It is in fact apparent from the calculations the importance of multiquan-tum processes.� 2007 Elsevier B.V. All rights reserved.

1. Introduction

The relaxation of vibrationally excited O2 molecules byatomic oxygen is a process of paramount importance indetermining the vibrational content of O2 under non-equi-librium conditions met in hypersonic flows, plasma chemis-try, plasma physics and atmospheric processes. Theaccepted idea, based on old theoretical and experimentalworks [1–6], is that atomic oxygen strongly deactivatesthe vibrationally excited molecules through the so-calledV–T process

O2ðv; jÞ þO ! O2ðw; j0Þ þO ð1Þwhere v, j and w, j 0 represent initial and final vibrationaland rotational quantum numbers, respectively. Anothercharacteristic of the process is the role of multiquantumtransitions in affecting the vibrational deactivation process.V–T rates for open-shell atom–molecule interaction includeboth reactive and non-reactive processes, thus requiring theuse of a reactive PES. CARS (coherent antistokes Ramanspectroscopy) has been used to measure the vibrational dis-tribution of oxygen molecules under glow discharge condi-

0009-2614/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2007.06.099

* Corresponding author. Fax: +39 080 5929520.E-mail address: fabrizio.esposito@ba.imip.cnr.it (F. Esposito).

tions. These measurements showed vibrationaldistributions close to Boltzmann ones at gas temperatureconfirming the deactivating action of atomic oxygen inthe discharge [7].

Accurate experimental work [8] has been recentlydevoted to the determination of the low temperature(T = 315 K) removal rate, i.e. to the rate constant of theprocess

O2ðv ¼ 1Þ þO ! O2ðw 6¼ 1Þ þO ð2ÞQCT (quasiclassical trajectory) calculations on a reliableDMBE PES (potential energy surface) of Varandas andPais [9] have been performed some years ago by Lagana’et al. [10–12]. These rates, while strongly improving theold QCT rates of Breig [3], cover only the vibrational quan-tum number range 5 6 v 6 30 and cannot be comparedwith the bulk of the experimental rate K10 [4,5] which referto process (3):

O2ðv ¼ 1Þ þO ! O2ðw ¼ 0Þ þO ð3ÞQCT rates for the dissociation process

O2ðv; jÞ þO ! 3O ð4Þhave been obtained by Esposito and Capitelli [13] using theDMBE PES. These rates cover the whole vibrational range0 6 v 6 46 and have been validated with global experimen-

1e–12

2e–12

3e–12

4e–12

5e–12

6e–12

7e–12

8e–12

9e–12

1e–11

1.1e–11

100 200 300 400 500 600 700 800 900 1000

rate

coe

ffici

ent (

cm3 /s

)

temperature (K)

v=1,w≠1 reactive var.factorv=1,w≠1 nonreactive var.factor

v=1,w≠1 total var.factorv=1,w≠1 total fixed factor

Kalogerakis et al.

Fig. 1. Comparison of vibrational removal rate coefficient from v = 1 as afunction of temperature calculated by quasiclassical method (presentwork) and experimentally (Kalogerakis et al.). Reactive, non-reactive andtotal computed rates are shown. For the total rates also the result withfixed degeneracy factor is presented.

F. Esposito, M. Capitelli / Chemical Physics Letters 443 (2007) 222–226 223

tal dissociation rates. The same code has been run forobtaining the complete set of VT rates. Some of these ratesare compared in this Letter with old and new experimentalVT rates. Note that the lacking of rotational numbers inthe notation means that a rotational temperature equalto the translational one is applied, and final rotation issummed up for each given final vibrational state.

2. Method of calculation

We have calculated cross sections for oxygen atom–mol-ecule collisions from and to each rovibrational state of thediatomic molecule considered. Given the huge amount ofcross sections to be computed, it has been necessary touse a quite efficient and fast method for dynamical calcula-tions. In these last years we have produced some works onatom–molecule collision processes using quasiclassicalapproach, finding results that appear in satisfying agree-ment with global experimental data [13–15]. The code forQCT has been entirely developed in our group, paying par-ticular attention in numerical error evaluation and timestep correction in trajectory computations (see Ref. [16]for some details). In this context we found it is essentiallyuncorrect to guess that a small (1%–5% typically) but notnegligible error in trajectory calculations can be in someway obscured by the large mass of results. This can be cor-rect if cross sections are acceptable in specific conditions‘favourable’ from a physical and numerical point of view(i.e. when probability of the event is high). But these con-ditions are not met in general when dealing with the wholeladder of diatomic rovibrational initial and final states. Theresults that we show in the following section are allobtained in this framework, with high numerical accuracyin trajectory calculations [16], with strict trajectory check-ing (not one in ten as usually), with continuous distributionof translational energy from 10�3 to 3 eV. The whole rovi-brational ladder has been calculated from the diatomic sec-tion of the PES by using WKB approximation, which isconsistent with product analysis in QCT calculations.Cross sections from each initial (v, j) state of the diatomicmolecule have been individually calculated, giving us thepossibility of constructing rate coefficients with any valueof vibrational and rotational temperatures, feature essen-tial for studying non-equilibrium systems. The relaxationrate coefficients obtained by cross sections have been trea-ted in two different ways: dividing the rate coefficient by thefixed degeneracy factor 27, already explained in Ref. [9],and using a variable degeneracy factor proposed in Ref.[17].

3. Results

Fig. 1 shows our total calculated removal K1w 6¼1 ratecoefficient as a function of temperature in the range100–1000 K calculated with fixed and variable degeneracyfactor. In the same figure we also report the non-reactiveand reactive contribution, obtained with variable statistical

factor. The first observation is that both reactive and non-reactive contributions are important in establishing therates, the reactive one being higher at low temperature.This behaviour is mainly due to the lack of reaction barrierin the used O3 PES. In the same figure we report the exper-imental removal rate of Kalogerakis et al. [8] at 315 K,obtained by measuring with tunable 193 nm laser lightthe concentration of O2(v = 1) level, formed by photolysis.

A difference up to about a factor 2 can be observedbetween our calculated value and corresponding experi-mental one; this difference decreases when fixed degeneracyfactor is used.

It is worth noticing the smooth minimum in the K1w 6¼1

plot into a temperature range of 550–750 K, dependingon the use of the degeneracy factor, but essentially indepen-dent of the exchange or non-reactive process considered.

Unfortunately no other experimental data are availablein the low temperature regime 100 6 T 6 1000. However,Kalogerakis et al. on the basis of available experimentsand theoretical calculations of the temperature dependenceof the isotope exchange reaction, suggest a possibleincrease of the relaxation rate of approximately 30% asthe temperature decreases from 300 to 230 K. Keeping inmind the original idea of Bauer and Tsang [2] that thevibrational relaxation of molecules by parent atoms occursentirely via exchange reaction (partially confirmed in thiswork), we compare our calculated exchange rate:

OþO2ðv ¼ 0Þ ! O2 þO ð5Þwith the corresponding experimental isotope exchangereaction values obtained by Fleurat-Lessard et al. [18] inthe temperature range 233–353 K (see Fig. 2). The cases(a) and (b) shown in Fig. 2 refer respectively to the endo-thermal exchange reaction from 16O + 18O18O, and to theexothermal case from 18O + 16O16O. In our case the iso-tope is invariably 16O, therefore an intermediate value

2e-12

2.5e-12

3e-12

3.5e-12

4e-12

4.5e-12

5e-12

5.5e-12

6e-12

100 150 200 250 300 350 400 450 500

rate

coe

ffici

ent (

cm3 /s

)

temperature (K)

present work with fixed factorpresent work with variable factor

Fleurat-Lessard et al.(a)Fleurat-Lessard et al.(b)

Fig. 2. Comparison of exchange rate coefficient as a function oftemperature calculated by quasiclassical method (present work) withisotope exchange obtained experimentally in the range 233–353 K(Fleurat-Lessard et al.). Theoretical cases are treated with a fixeddegeneracy factor of 1/27, or with a variable factor described in the text.Experimental case (a) refers to endothermal exchange reaction from16O + 18O18O, while case (b) refers to exothermal 18O + 16O 16O.

224 F. Esposito, M. Capitelli / Chemical Physics Letters 443 (2007) 222–226

should be considered for this approximate comparison. Itis clear that the isotope effect is comparable to the theoret-ical incertitude in assigning the correct degeneracy to theseresults, which is linked finally to the unavailability of de-tailed information of possible non-adiabatic transitions inthe studied reaction. The trends with temperature of bothexperimental and theoretical rates are clearly decreasing.

Fig. 3 shows the experimental K10 data of Kiefer andLutz covering the temperature range 1600 K 6 T 6 3300 Kand of Breen et al. in the temperature range1000 K 6 T 6 3400 K obtained by conventional shocktube measurements. The Ps experimental data have beentransformed in K10 using the following equation:

Ps ¼ kBT=K10 ð6Þ

5e-12

1e-11

1.5e-11

2e-11

2.5e-11

500 1000 1500 2000 2500 3000 3500 4000

rate

coe

ffici

ent (

cm3 /s

)

temperature (K)

v=1,w=0 total var.factorv=1,w≠1 total var.factor

Breen et al.Kiefer Lutz

Fig. 3. Comparison of vibrational deactivation rate coefficient from v = 1to w = 0 obtained experimentally (Breen et al., Kiefer and Lutz) andcomputationally in this work. Also our removal rate coefficient v = 1 tow 6¼ 1 is shown, with variable degeneracy factor.

based on the simple relaxation of level 1 considered as aminor species compared to the ground state oxygen mole-cules (kB is the Boltzmann constant). In the same figurewe are reporting our calculated K10 values, as well as thetotal removal rate K1w 6¼1 from v = 1, with variable degener-acy factor.

First of all we would like to underline that the Breenet al. experimental data have about the same absolute valueas those of Kiefer and Lutz, but display less temperaturedependence. In the same temperature range our K10 valuesare smaller than the experimental results by factor in therange 1–5, exhibiting a weaker temperature dependence.A possible explanation of this difference is linked to thedeconvolution of the experimental data, which completelydisregards the multiquantum transitions. This point canbe better understood by inspection of our removal rateK1w 6¼1 in the same figure, which of course includes the effectof multiquantum transitions.

As a whole the comparison of our calculated K1w 6¼1 andK10 values with available experimental results indicates afair amount of confidence in the whole temperature rangeexamined. This suggests that the extrapolation of high tem-perature results to low temperature, largely used in the lit-erature, leads to a strongly underestimation of the actualvalues.

In this context we discuss relationships between hightemperature Ps and K10 values. Results reported in Fig. 3have been obtained using Eq. (6). However, other formula-tions have been proposed in the literature. The most popu-lar one is due to Landau and Teller who proposed thefollowing equation (see for example, Ref. [19],[20])

Ps ¼ kBT=½K10ð1� expð�E=kBT ÞÞ� ð7Þwhere E/kB is the characteristic vibrational temperature.This equation is derived by the vibrational kineticswhich considers the levels as harmonic and affected onlyby monoquantum transitions with the scaling lawKv,v�1 = v K10.

Another formulation, based on the kinetics of 0,1 levels,is given by Levine and Bernstein [21]

Ps ¼ kBT=½K10ð1þ expð�E=kBT ÞÞ� ð8ÞEq. (8) differs from Eq. (6) for the inclusion of the reverseprocess of Eq. (2).

Fig. 4 shows the Ps values calculated from our QCTresults using the three formulations as in Eqs. (6)–(8).As expected the three formulations give different Ps val-ues from a given temperature on (about 800 K in the pres-ent case). It should be again stressed that the Psformulation due to Landau and Teller is a rough approx-imation when multiquantum transitions are important inrelaxing the vibrational content of the molecules (seebelow).

This point can be understood by inspecting Fig. 5 wherethe multiquantum QCT kvw rates calculated at T = 300 Kare shown. We are considering the rate coefficients of theprocess:

1e-09

1e-08

1e-07

1e-06

100 200 400 800 1600 3200

Pτ

(at

m s

)

temperature (K)

eq.6eq.7eq.8

Fig. 4. Ps product as a function of temperature deduced from our QCTresults using Eqs. (6)–(8).

1e-14

1e-13

1e-12

1e-11

0 5 10 15 20 25 30 35 40 45 50

rate

coe

ffici

ent (

cm3 /s

)

initial vibrational state

Δv=1Δv=5

Δv=10Δv=20Δv=30

Webster and Bair

Fig. 5. Calculated multiquantum de-excitation rate coefficients as afunction of initial vibrational quantum number, for fixed quantumnumber jumps Dv, at T = 300 K. Comparison with experimental datafrom Webster and Bair.

F. Esposito, M. Capitelli / Chemical Physics Letters 443 (2007) 222–226 225

OþO2ðvÞ ¼ OþO2ðwÞ ð9Þas a function of the initial vibrational quantum number v

and for different Dv = v–w values. We can see that the largemajority of multiquantum transitions are independent ofthe initial vibrational quantum number and quantumjumps. Deviation from this behaviour are observed forv < 6 and v > 30. We can note infact that the monoquan-tum v! v – 1 transition rates decrease up to v = 6, present-ing approximately a flat profile from v = 6 to v = 30. Then,we observe an increase with v up to a maximum at v = 42followed by a sharp decrease, due to the opening of the dis-sociation channel. A similar behaviour is presented for themultiquantum transitions. In the same figure we have alsoreported a fit to experimental results obtained by Websterand Bair for relaxation of high vibrational excited O2(v)molecules formed by flash photolysis of O2–Ar mixtures.

The experimental data were reproduced by using the fol-lowing equation

Kvw ¼ K10½1þ 0:04 � ðv� 1Þ� ð10Þwith K10 = 10�12 cm3 s�1 taken from the extrapolation ofthe experimental data of Kiefer and Lutz. It should benoted that the experimental values are independent of thefinal vibrational state (they only linearly depend on theinitial vibrational state) and particularly refer to the vibra-tional range 14 6 v 6 19. The agreement between theoreti-cal and experimental results is indeed satisfactory if onetakes into account the approximate nature of the experi-mental results and the difficulty of the present calculations.

4. Conclusions

We have reported a complete QCT study of the relaxa-tion of oxygen vibrationally excited molecules by atomicoxygen. The results, while confirming the old findings onthe effectiveness of VT relaxation and the importance ofmultiquantum transitions, show a satisfactory agreementwith the existing experimental and theoretical results. Theyalso prompt new experimental results for the high temper-ature range in order to eliminate their dependence on theassumed Landau Teller kinetic model. Another importantfeature of the calculated values is the high similarity ofinelastic and exchange rates, contrarily to the higherimportance commonly attributed to the reactive part forthis system. Improvements of the present results can beobtained by using more accurate PES as in [22,23], as wellas by estimating in a more reliable way the role of the man-ifold of potential surfaces correlating with Oð3PÞþO2 X 3

P�g

� �system.

As a final comment we want to point out the importanceof the rates discussed in this Letter for future modeling ofthe vibrational relaxation of oxygen molecules underplasma [24] and atmospheric conditions [25], often ratio-nalized in terms of pure molecule–molecule interactions[25,26].

Acknowledgments

This work has been partially supported by MIUR FIRBProject No. RBAU01H8FW003 ‘‘Dinamica microscopicadella reattivita’ chimica’’ and MIUR PRIN 2005 ProjectNo. 2005039049005 ‘‘Dinamica Molecolare di processi ele-mentari in condizioni di non equilibrio’’.

References

[1] M. Cacciatore, M. Capitelli, M. Dilonardo, Beit. Plasmaphys. 18(1978) 279.

[2] S.H. Bauer, S.C. Tsang, Mechanisms for vibrational relaxation athigh temperatures, Phys. Fluids 6 (1963) 182.

[3] E.L. Breig, J. Chem. Phys. 51 (1969) 4539.[4] J.H. Kiefer, R.W. Lutz, The Effect of Oxygen Atoms on the

Vibrational Relaxation of Oxygen, 11th Symp. Int. on Combustion,1967, p. 67.

[5] J.E. Breen, R.B. Quy, G.P. Glass, J. Chem. Phys. 59 (1973) 557.[6] H. Webster, E.J. Bair, J. Chem. Phys. 56 (1972) 6104.

226 F. Esposito, M. Capitelli / Chemical Physics Letters 443 (2007) 222–226

[7] M. Baeva, X. Luo, B. Pfelzer, T. Repsilber, J. Uhlenbusch, PlasmaSources Sci. Technol. 9 (2000) 128.

[8] K.S. Kalogerakis, R.A. Copeland, T.G. Slanger, J. Chem. Phys. 123(2005) 194303.

[9] A.J.C. Varandas, A.A.C.C. Pais, Mol. Phys. 65 (1988) 843.[10] A. Lagana’, E. Garcia, T. Martinez, Quasiclassical and Quantum

Rate Coefficients for the O + O2 Reaction, Chemistry Department,University of Perugia, Italy, 1998.

[11] A. Lagana’, A. Riganelli, G. Ochoa De Aspuru, E. Garcia, MT.Martinez, Chem. Phys. Lett. 288 (1998) 616.

[12] A. Lagana’, S. Crocchianti, G. Ochoa de Aspuru, A. Riganelli, E.Garcia, Plasma Sources Sci. Technol. 6 (1997) 270.

[13] F. Esposito, M. Capitelli, Chem. Phys. Lett. 364 (2002) 180.[14] F. Esposito, PhD Thesis, Bari University, Bari, Italy, 1999.[15] F. Esposito, I. Armenise, M. Capitelli, Chem. Phys. 331 (2006) 1.[16] F. Esposito, M. Capitelli, Chem. Phys. Lett. 418 (2006) 581.[17] A. Gross, G.D. Billing, Chem. Phys. 217 (1998) 1.[18] P. Fleurat-Lessard, S. Yu. Grebenshchikov, R. Schinke, C. Janssen,

D. Krankowsky, J. Chem. Phys. 119 (2003) 4700.

[19] C.F. Hansen, Rate Processes in Gas Phase, NASA ReferencePublication 1090 (1983), p. 99 (available from the NASA web site).

[20] G.G. Chernyi, S.A. Losev, S.O. Macheret, B.V. Potapkin, in: PaulZarchan (Ed.), Physical and chemical processes in gas dynamics: crosssections and rate constants, Progress in Astronautics and Aeronau-tics, 196, 2002, p. 85, on the first volume.

[21] R.D. Levine, R.B. Bernstein, Molecular Reaction Dynamics, OxfordUniversity Press, 1974, chapter 5, p. 127.

[22] R. Siebert, R. Schinke, M. Bittererova, Phys. Chem. Chem. Phys. 3(2001) 1795.

[23] D. Babikov, B.K. Kendrick, R.B. Walker, R.T. Pack, P. Fleurat-Lessar, R. Schinke, J. Chem. Phys. 118 (2003) 6298.

[24] M. Capitelli, C.M. Ferreira, B.F. Gordiets, A.I. Osipov, PlasmaKinetics in Atmospheric Gases, Springer Series on Atomic, Opticaland Plasma Physics, Springer-Verlag, Berlin, Heidelberg, 2000.

[25] R. Atkinson et al., Atmos. Chem. Phys. 4 (2004) 1461.[26] C. Coletti, G.D. Billing, Chem. Phys. Lett. 356 (2002) 14.