The Ozone Isotope Effect. Answers and Questions

The Ozone Isotope Effect Answers and Questions

The Ozone Isotope Effect Answers and Questions Dynamical studies of the ozone isotope effect: A status report Ann. Rev. Phys. Chem. 57, 625 661 (2006) R. Schinke S.Yu. Grebenshchikov, M. V. Ivanov and P. Fleurat-Lessard

Some basic facts about O, O 2 and O 3 isotopes of oxygen: 16 O (0.99763), 17 O (3.7 10 4 ), 18 O (2.0 10 3 ) 6 7 8 zero point energies (ZPE) of O 2 : E ZPE ω/2 ω f/µ µ = m 1 m 2 /(m 1 +m 2 ) 66: 790.4cm 1 68: -22.2cm 1 (1eV = 8066cm 1 ) 88: -45.2cm 1 (for comparison: k B T = 220cm 1 at 300K)

forms of ozone: ozone is predicted by theory to exist in two different forms: R1 R2 D 3h, cyclic O3 C 2v, open O3 = 60 = 117 However, only Open Ozone exists in the gas phase; the central atom is special: 687 6+87 or 68+7 E ZPE (87) < E ZPE (68) 687 67+8 is not possible

Some historical remarks about O 3 isotope effect 1981: Mauersberger measures the fractionation δ( 50 O 3 ) 13% (heavy Ozone 668) in the stratosphere (balloon experiments) 1985: Thiemens measures the fractionation δ( 49 O 3 ) 11% (667) in laboratory experiments Mauersberger et al. Adv. At. Mol. 1990: more laboratory experiments Opt. Phys. 50, 1 54 (2005) δ( M O 3 ) = [ ] ( M O3 / 48 O 3 ) meas. 1 ( M O 3 / 48 O 3 ) cal. 100 very large enrichments no apparent mass dependence δ( 49 O 3 ) δ( 50 O 3 ) ozone isotope effect or ozone anomaly

Ozone recombination or formation rate constants Ozone formation rate: d[o 3 ] dt = k rec (T)[O][O 2 ][M] [O] [O 2 ] [M] Mauersberger and coworkers measured (under controlled conditions in the laboratory) k rec for several [O,O 2 ] combinations (relative to 666): 6+66: k rec = 1.00 (normalization) 6+88: k rec = 1.50 (largest ratio) 8+66: k rec = 0.92 (smallest ratio) 6+68: k rec = 1.45 etc. The measured k rec /k 666 show a large variation with no apparent systematic dependence...

... until they were represented as function of the ZPE difference between the two possible diatomic channels: ZPE = E ZPE (products) E ZPE (reactants) exothermic endothermic 8 + 66 866 86 + 6 + 23 cm 6 + 88 688 68 + 8-23 cm -1-1 Janssen et al. (2001) The symmetric molecules behave differently than the non-symmetric ones! Symmetric 666, 868 etc.

The fractionation constants follow from the recombination rate constants k rec Therefore, the k rec are the focus of most theoretical studies

Recombination vs. isotope exchange reaction (1) O+PQ (OPQ) formation of highly excited complex (2) (OPQ) O+PQ inelastic process (e.g., vib. relaxation) (OPQ) OP+Q isotope exchange (3) (OPQ) +M OPQ+M stabilization (energy transf. mechanism) relaxation, isotope exchange and recombination are intimately related: they proceed through the same O 3 complex. reactions (2) are well defined (bi-molecular collisions) and can be rigorously treated; they are independent of pressure p. stabilization step (3) involves many collisions with M and is extremely complicated to treat (for example, master equation); it shows a strong p dependence. at low pressures: isotope exchange is much faster than stabilization

O+O 2 O 3 interaction potential first reasonable potential energy surface (PES) calculated by Siebert et al. in 2001 and 2002 multi reference configuration interaction (MRCI) cc-pvqz basis set global PES V(R 1,R 2,α) R 1 and R 2 are the two O O O bond lengths, α is the angle

2D contour representations 150 α [deg.] R 2 [a 0 ] 120 90 60 30 5 4 3 cyclic O 3 E [ev] 1.0 0.5 0.0 3 4 5 6 R 1 [a 0 ] V(R 1,R 2,α) three equivalent wells + cyclic well accurate vibrational energies very small dissociation barrier (0.006 ev) narrow transition state quite harmonic, compact potential (correlation with excited products?) 2 2 3 4 5 R 1 [a 0 ]

1.10 II 0.014 ev bettercalculationsincreased e and decrease the barrier! simple modification = potential II E [ev] 1.00 III +0.006 ev I artificial removal of barrier = potential III 0.90 0.80 3 4 5 6 7 8 9 R 1 [a 0 ]

15 I 10 σ [a 0 2 ] 5 0 15 II exchange reaction O+O 2 (j) O 2 (j )+O σ [a 0 2 ] 10 classical trajectory calculations 5 0 initial state resolved cross sections for isotopic exchange σ [a 0 2 ] 40 30 20 j=0 j=10 j=20 j=40 III σ ex j (E coll.) depend strongly on the transition state barrier! 10 0 0 1000 2000 3000 E c [cm -1 ]

Exchange reaction rate constant k ex (T) exp. artificial PES (III) original PES (II) poor agreement with experimental rate quantum effects unlikely for three heavy O atoms PES (transition state) much better (i.e., more expensive) calc. do not change the TS structure non-adiabatic effects, i.e. breakdown of BO approximation?

New (2004) ab initio calculations in the transition-state region at our computational limit 180 160 140 +100 0 AQCC, av6z basis set 120 [deg] 100 80 r R not a full PES (R O2 = r fixed) 60 40 20 0-200 R [ a 0] -100 4 4.5 5 5.5 6 6.5 7 7.5 8 [see also: Holka et al., J.Phys.Chem. A 36, 9927 (2010)] The structure of a narrow TS with the barrier below the asymptote is confirmed!

Non-adiabatic transitions between different electronic states all correlating with O( 3 P)+O 2 (X 3 Σ g ) (open shell system) (a) E [cm -1 ] S T Q 3 (5 + 3 + 1) = 27 different electronic states correlate with the ground state asymptote. E [cm -1 ] j=0 j=1 (b) Thus, transitions due to non-adiabatic, spin-orbit or Renner-Teller coupling are possible! j=2 spin-orbit splitting R [a ] 0

Isotope dependence of exchange reaction the ratio R 8,6 = k 8+66 86+6 k 6+88 68+8 has been measured (directly) it is 1.27 at room temperature ( ZPE = ±23cm 1 2k B T = 440cm 1 ) 8 + 66 (866)* 86 + 6 6 + 88 (688)* 68 + 8 ZPE ZPE exothermic endothermic

Quantum mechanics automatically includes ZPE classical mechanics, however, does not! simple trick: we add ZPE to V(R 1,R 2,α) in the asymptotic channels (thereby making the PES mass-dependent). 0-200 E [cm -1 ] -400-600 -800 4 5 6 7 R [a.u.] O 3 original PES O 3 PES + ZPE

the classical method ( ) with mass-dependent PES works well; slight underestimation of ratio R 8,6 another classical method ( ) gives even better results; it is, however, much more expensive (about 95% of trajectories are not counted)

Recombination within the strong-collision model deactivation and activation of the excited complex in multiple collisions with M is very difficult to describe. strong-collision model: stabilization occurs in a single collision with frequency ω, which is the sole parameter! ω p and ω E/collision for each trajectory (i) we define a stabilization probability P (i) stab = 1 e ωτ i low-pressure limit: P (i) stab ωτ i high-pressure limit: P (i) stab 1 τ i = survival time of complex linear p dependence every complex-forming trajectory is stabilized

pressure dependence of recombination rate k rec k stab (p) [cm 3 s -1 ] 10 10 10 10-11 -12-13 -14 10 T=300K -1 [ps ] -5-4 -3 10-2 10 10 Hippler et al. Lin and Leu p = 7 10 23 ω [p] =molec./cm 3 [ω] =ps 1 the high-p behaviour is not understood! 10-15 18 10 10 19 20 10 10 21 p [molec. cm -3 ] 10 22

temperature dependence of recombination rate k rec 10-32 ENERGY TRANSFER CHAPERON (a) (b) k r / [Ar] [cm 6 molecule -2 s -1 ] 10-33 10-34 10-35 100 1000 T [K] 100 1000 T [K] ET mechanism yields T dependence, which is too weak at lower T

Recombination within the chaperon model The chaperon mechanism is a one-step process (J. Troe): Ar O+O 2 O 3 +Ar Ar O 2 +O O 3 +Ar, where Ar O and Ar O 2 are weakly bound vdw dimers. k r,ch (T) K ArO (T) k ArO+O2 O 3 +Ar(T) [M] where K ArO is the equilibrium constant of the Ar+O Ar O system. Both, k ArO+O2 O 3 +Ar and K ArO strongly depend on T.

temperature dependence of recombination rate k rec 10-32 ENERGY TRANSFER CHAPERON (a) (b) k r / [Ar] [cm 6 molecule -2 s -1 ] 10-33 10-34 Troe et al. 10-35 100 1000 T [K] 100 1000 T [K] Chaperon mechanism yields reasonable T dependence at lower T. However, is it really a one-step mechanism?

Isotope dependence of recombination rate at low pressures: k rec ω τ aver. 8 + 66 (866)* 86 + 6 6 + 88 (688)* 68 + 8 ZPE ZPE exothermic endothermic smaller τ aver. smaller k rec larger τ aver. larger k rec k rec = 0.92 k rec = 1.50

comparison of exp. and calculated recombination rate coefficients exothermic endothermic 8 + 66 866 86 + 6 + 23 cm 6 + 88 688 68 + 8-23 cm -1-1 Symmetric norma. 666, 868 etc. the overall dependence is well reproduced by the classical calculations

comparison of exp. and calculated recombination rate coefficients exothermic endothermic 8 + 66 866 86 + 6 + 23 cm 6 + 88 688 68 + 8-23 cm -1-1 15% Symmetric norma. 666, 868 etc. the overall dependence is well reproduced by the classical calculations when ZPE is included! however, the rates for the symmetric molecules are too high by about 15%

Classical vs. statistical (RRKM) calculations The classical results for the isotope dependence agree with the statistical (RRKM) results of Marcus et al. (1999 2002) They agree because in both approaches ZPE is included. Otherwise, the two methods are quite different! Marcus et al. introduced a so-called non-statistical parameter η 1.18 in order to (artificially) decrease the rates for the symmetric molecules. With η = 1 very poor results for measured fractionations (Marcus)! Up to now, there is no computational verification nor a real understanding of this rescaling!

Is the O+O 2 O 3 statistical? low density of states near dissociation threshold (ρ 0.1 per cm 1 ) shape of wave functions, assignability even close to threshold slow intramolecular rotational-vibrational energy transfer (see below) molecular beam experiment at 0.32eV collision energy for the O+O 2 exchange reaction shows a clear forward backward asymmetry (Van Wyngarden et al. J. Am. Chem. Soc. 129, 2866 (2007) exact quantum mechanical calculations for collision energies as low as 0.01 0.05eV and j = 0 also show clear forward backward asymmetry (Sun et al. PNAS 107, 555 (2010))

40 30 Classical Statistical I Comparison between classical and statistical σ(e coll.,j ) σ [a 0 2 ] 20 10 0 j=0 j=20 the state-specific statistical cross sections are very different from the classical ones! 100 80 j=0 III the dependence on E c and j is very different σ [a 0 2 ] 60 40 however, the averaged rate constants are similar what does that mean? 20 j=20 0 0 200 400 600 800 E c [cm -1 ]

Need for quantum mechanical calculations classical(as well as statistical) calculations are questionable at very low energies the difference between symmetric and non-symmetric O 3 strongly indicates that the symmetry of the quantum states is important in quantum mechanics (schematic): ( ) ĥsym 0 Ĥ sym = 0 ĥ anti sym Hamiltonian block-diagonal wavefunctions are either symmetric or anti-symmetric, without any coupling between the two sets this may affect the energy flow in O 3 and thus τ aver. and/or ω E coll symmetry is not included in classical mechanics nor in the statistical approach

Quantum mechanical resonances resonances are the continuation of the true bound states into the continuum 10 2 10 1 original PES (2001) E res = E 0 iγ/2 lifetime = τ = Γ 1 Γ [cm -1 ] 10 0 10-1 (0,12,0) (8,0,0) S.Yu. Grebenshchikov, R. Schinke, and W.L. Hase In Comprehensive Chemical Kinetics, Vol. 39 10-2 10-3 0 200 400 600 800 1000 E - E thres [cm -1 ]

quantum mechanical resonances (J = 0) Babikov et al. (2003) 18 What are the very long-lived states between the two thresholds (shaded area)?

S. Yu. Grebenshchikov O 3 The long-lived resonances between thresholds are the vdw states in the upper channel 8 66. Decay only by coupling to the main O 3 well and subsequently to the continuum of the other vdw well 6 86, i.e., they are almost real bound states. Do such delocalized vdw states contribute to the recombination???

most complete quantum mechanical calculations up to now k rec (T) = Q 1 r JK(2J +1) n Γ n (JK)ω ω +Γ n (JK) e E n(jk)/k b T resonance energies E n (JK) and widths Γ n (JK) for J 40 and K 10 (several thousand!!) simplified PES: no vdw wells and only one (rather than three) O 3 well results presented in next talk!

Vibrational energy transfer in O 3+Ar collisions classical trajectory calculations problem: separation of vibrational and active rotational (K a ) energy maximum impact parameter; what is a collision? infinite order sudden approximation quantum mechanical approximation, full PES τ coll τ rot breathing sphere approximation drastic quantum mechanical approximation average full 6D PES over Ar O 3 orientations = 4D PES preserves symmetry!

10 0 Ivanov et al. Mol. Phys. 108, 259 (2010) 10-1 IOSA black: 668 (non-symmetric) - E [cm -1 ] 10-2 BSA red: 686 (symmetric) trajectory and IOS calculations agree well 10-3 no apparent difference between symmetric and non-symmetric O 3 10-4 -6000-4000 -2000 0 E [cm -1 ] E vib 0.5 1cm 1 near threshold

Other approach to collisional energy transfer: Ivanov and Babikov (Tuesday afternoon)

Intramolecular vibrational rotational energy flow classical trajectory calculations, E int E threshold : higly excited ozone E int = E rot (t)+e vib (t) = constant E rot (t) = AK 2 a +BK 2 b +CK2 c K x projection of J on body-fixed x-axis J = constant E vib E rot energy flow (Coriolis coupling) magnitude and direction depend strongly on K a similar calculations (with similar results) by Kryvohuz and Marcus: J.Chem.Phys. 132, 224304 and 224305 (2010)

T r = - E v [cm -1 ] 60 40 20 0-20 -40-60 -80-100 -120 K a (0)=2 K a (0)=6 K a (0)=10 K a (0)=14 K a (0)=18 low K a high K a 0 100 200 300 400 t [ps] low K a : flow from vibration to rotation high K a : flow from rotation to vibration possible mechanism of stabilization: 1. flow of energy from vib. to rot. during collisions with M 2. removal of rot. energy in collisions with M

Open Questions magnitude and T dependence of k ex? T dependence of k recom? transition-state ( reef ) structure of PES is essential

dynamical-weighting state-averaged CASSCF orbitals up to 10 excited 1 A states included smooth change of orbitals through reef region

Open Questions magnitude and T dependence of k ex? T dependence of k recom? transition-state ( reef ) structure of PES is essential magnitude of energy transfer per collision with M? (1cm 1 vs. 10cm 1 ) quantum mechanical test of intramolecular V R energy transfer why are symmetric and non-symmetric isotopomers formed with different rates (η 1.15)? different rates of intramolecular V R energy transfer for sym. and non-sym. complexes?

... or something else has been ignored: presentation by P. Reinhardt and F. Robert (Tuesday afternoon)