Chemical Physics Letters 372 (2003) 686–691 www.elsevier.com/locate/cplett Quantum origin of an anomalous isotope effect in ozone formation D. Babikov *, B.K. Kendrick, R.B. Walker, R. Schinke 1, R.T Pack Theoretical Division, Mail Stop B268, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 2 January 2003; in final form 14 February 2003 Abstract Accurate quantum mechanical calculations of the J 0 energies and lifetimes of the metastable states of ozone on ð ¼ Þ a new, accurate potential energy surface are reported. These are very relevant to a famous anomalous isotope effect in the reaction that forms ozone because of their role in the energy transfer mechanism, in which metastable ro-vibrational states of ozone are formed and then stabilized by collisions with a third body. The resulting spectrum of metastable states is very dense below the delta zero-point energy threshold and sparse above it. This gives a clear qualitative explanation of why the effect occurs. Ó 2003 Elsevier Science B.V. All rights reserved. 1. Introduction independentÕ. Because of its importance for at- mospheric chemistry, planetary science, and fun- The origin of the anomalously large enrich- damental reaction dynamics, this phenomenon has ments of stratospheric ozone in the heavy isotopes been intensively studied (for recent short reviews, of oxygen has been a mystery for 20 years [1,2]. see Thiemens [3] and Mauersberger et al. [4]). Oxygen has three stable isotopes: 16O, 17O and 18O. For several years, it was not at all clear as to The isotope 16O is dominant in the atmosphere, so which step in the atmospheric ozone cycle was 16 that most oxygen molecules, O2, only include O isotopically selective, but careful experimental atoms. However, stratospheric ozone, O3, is sur- work [5] traced it clearly to the recombination re- prisingly observed to be heavily 10% enriched action that forms ozone, 17 18 ð$ Þ in O and O relative to the background O2 O O2 M O3 M R1 present. Further, the enrichment is about equal in þ þ ) þ ð Þ 17O and 18O, which led to its being called [2] Ômass Here the third body M is any atom or molecule that can carry away the excess energy. The product 16 17 O3 may include any combination of O, O and 18O isotopes. Mauersberger and co-workers [6–9] * Corresponding author. Fax: 1-505-665-3903. E-mail address: [email protected] (D. Babikov). have measured the rate coefficients for reaction 1 Present address: Max-Planck Institut fuur€ Stroomungsfor-€ (R1) for almost all of the possible isotopic com- schung, D-37073 Goottingen,€ Germany. binations of reactants O and O2. The rates do 0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00479-2 D. Babikov et al. / Chemical Physics Letters 372 (2003) 686–691 687 depend on the masses of the oxygen atoms in- hand side of this reaction, and reaction (R2) is part volved, and, for different isotopic compositions the of this reaction. When different isotopes of oxygen rates differ by more than 50%, which is a re- (x; y and z) are involved, the zero-point energies of markably large isotope effect! the O2 molecules on the right- and left-hand sides The observed isotope effect in reaction (R1) is may be different, and reaction (R4) can thus be essentially independent [10,11] of the identity of slightly exothermic or endothermic and exhibit the third body M, and occurs with M being as slightly different rates. The Mauersberger group simple as Ar or He. For this simple M, the reaction [7,8] speculated that Ôthe endothermic exchange mechanism is expected to be dominated by the process may lead to a longer-lived collisional energy transfer mechanism: complex which then has a higher probability to become stabilized to ozoneÕ. O O OÃ R2 þ 2 () 3 ð Þ In their next paper Hathorn and Marcus [13] further discussed the two factors. They showed that OÃ M O M R3 3 þ ) 3 þ ð Þ their nonstatistical empirical factor g needed to Here O3Ã is a ro-vibrational metastable state (i.e., have a value of about 1.15 but could not identify its scattering resonance) which lives long enough to origin. They also noted that the partitioning factors be stabilized by the collision with M. Again, any are related to, and the relative rates of reaction combination of O isotopes can be involved. (R1) correlate with, differences in ratios of reduced Early attempts to explain the isotope effects masses and rotational constants as well as the using simple symmetry arguments or classical DZPE of the oxygen molecules involved (see also mechanics had little success. However, recent [14]). They also saw that at low energies one exit progress has been rapid. Hathorn and Marcus [12] channel in reaction (R4) Ôis energetically excluded have formulated the problem using a statistical and so the excited molecule has a longer lifetimeÕ RRKM-based theory. In addition to the usual and a Ôgreater chance to be deactivatedÕ by colli- symmetry number ratio of 2, they introduced two sions. Then, a nearly simultaneous pair of papers other factors that could produce isotope effects. [15,16] clearly identified the DZPE in reaction (R4) The first of these is an empirical non-statistical as the dominant factor in controlling the lifetimes factor (g parameter) to describe a difference in of the O3Ã and producing the large differences in the density of dynamically active O3Ã states in sym- relative rates of reaction (R1). The paper by Gao metric and non-symmetric ozone molecules. They and Marcus [15] was submitted first, but the paper demonstrated that this g effect is dominant in de- by the Mauersberger group [16] was published first. termining the anomalous mass-independent iso- We pause to remark that Gao and Marcus noted tope enrichments in stratospheric conditions. They in this paper [15] that their g factor might arise [12] also mentioned a second factor to explain the from collisional effects in reaction (R3). We believe mass-dependence observed in the recombination that to be correct and we shall discuss the idea a rates. That is a Ôpartitioning factorÕ that depends little farther in the discussion section. However, all on the Ôdifferences in zero-point energies of the two the calculations in Letter deal with the DZPE effect transition statesÕ in reaction (R2) that connect with in reaction (R2). Although the parameters in the isotopically different O2 molecules. In a paper semiempirical theory [15,17] have allowed the ex- published just a few weeks later, the Mauersberger perimental data for both ozone isotope effects to be group [7,8] noted that there is a correlation be- fit quite well, neither that theory [15,17] nor the tween the ozone formation rate coefficients and the experiments to date have been able to actually give zero-point energy change DZPE of the O mole- the spectrum of metastable states of ozone, and ð Þ 2 cules in the atom exchange reaction, neither group has been able to prove why such a x y z x y z x y z small thing as the DZPE should have such a large O O O O O O Ã O O O: R4 þ () ð Þ () þ ð Þ effect. Both groups have noted that accurate x y z Here, the metastable state O O O Ã, can be quantum mechanical calculations of positions and ð Þ formed from and decay to either the right- or left- lifetimes of the O3Ã states are required. 688 D. Babikov et al. / Chemical Physics Letters 372 (2003) 686–691 Charlo and Clary [18] have done approximate going and will be reported elsewhere. The lifetimes quantum calculations of the metastable states and of the metastable states [25] were obtained as a recombination rate and found isotopic effects of trace of SmithÕs Q-matrix. the right order of magnitude but often in the In this Letter calculations on two isotopic wrong direction. It is clear that, to accurately de- compositions of ozone are reported: 16O18O18O 16 16 18 scribe the O3Ã states for a molecule as complex as and O O O. The first of these is formed in the ozone, one must perform state-of-the-art calcula- following recombination reactions: tions. In this Letter we report rigorous quantum 16O 18O18O M 16O18O18O M R5 mechanical calculations of the energies and life- þ þ ) þ ð Þ times of the OÃ metastable states for total angular 16 18 18 16 18 18 3 O O O M O O O M R6 momentum J 0 on a new, accurate potential þ þ ) þ ð Þ energy surface.¼The results show the reason for the Experimentally, reactions (R5) and (R6) exhibit remarkable effect of the DZPE. very different rates: 1.50 and 0.92, respectively [16], relative to the reaction with all atoms being 16O. (The metastable states of the symmetric isotopomer 2. Calculations, results, and discussion 18O16O18O are also obtained in these calculations and will be discussed elsewhere.) For the second A very sophisticated potential energy surface isotopic combination, 16O16O18O, there are also two (PES) for ozone was used for this work. It is based entrance channels, 16O 16O18O and 16O16O 18O, þ þ on an accurate ab initio PES [19,20] calculated by exhibiting relative recombination rates similar to one of us. However, it also includes a correction in those above: 1.45 and 0.92, respectively [16]. the barrier region to make it agree with even more The low energy E < 110 K calculated lifetime ð Þ accurate ab initio calculations which we have spectra for the two isotopic combinations are performed along the minimum energy path [21,22].