1. Quantum Dynamics Studies of Anomalous Isotope Effects Dmitri Babikov Marquette University, Chemistry Department Milwaukee, Wisconsin, USA.

2. Outline What we learned about the MIF from studies of the anomalous isotope effect in O 3 ? What remains a challenge for theory and experiment on ozone? What has already been done on the S-containing species? What can be done in the near future on the S-containing species?

3. (Mauersberger, Geophys. Res. Lett. 8, 935, 1981) (Heidenreich & Thiemens, JCP 78, 892, 1983) In the laboratory studies: equal in 17 O and 18 O “mass independent” Discovery of Enrichments in O 3 In the atmosphere: “anomalous” O + O 2 + M  O 3 + M (Mauersberger and co., J. Geophys. Res. 95, D1, 901, 1990). Rates can differ by more then 50%, remarkably large isotope effect taking into account a small change of mass! Traced to the three-body recombination reaction which forms ozone:

4. First Round of Theoretical Research on Anomalous Isotope Effect YearMethodAuthorComments 1981Experimental DiscoveryMauersberger Stratosphere 1983StatisticsKaye, StrobelDepletion? 1986 - 1992 9 papersBatesFailed on rates 1996SIKIEGelleneContradicts exp. results 1997Classical TrajectoriesGross, Billing also Schatz Tiny effect, in wrong direction This is not a complete list…

5. State of the Problem before the Y 2000 “Currently, in spite of 15 years of intensive experimental and theoretical investigation,.. the mechanism that is responsible remains unidentified.” “Despite the progress that has been made during the past 10 years, a convincing physical explanation of the process that results in enrichment is still missing.” M. H. Thiemens, Science 283, 341 (1999) K. Mauersberger, Science 283, 370 (1999) - Theory was not particularly successful. - Experiments results were not complete and were not summarized in the form useful to theoreticians. - Situation changed dramatically in the XXI century!

6. Relative Rates (Exp.) (Mauersberger and co., PCCP 3, 4718, 2001) Very strong isotope effects. No any obvious correlation with O 3 masses.

7. Experimental Rates vs. DZPE 18→16 16→18 17→16 16→17 18→17 17→18 16→16 17→17 18→18 (Mauersberger and co., PCCP 3, 4718, 2001) x O + y O z O → ( x O y O z O ) * → x O y O + z O + M → O 3 Their explanation of the correlation: ZPEs of O 2 on the reactant and product sides are different. The atom exchange reaction is slightly exothermic or endothermic. This affects rate of the reaction, and lifetime of the intermediates. This paper stimulated new round of theoretical work.

8. YearMethodAuthorComments 1981Experimental DiscoveryMauersberger Stratosphere 2001 - 2004 RRKM + Non-statistical effect + Tuning parameters Gao, Marcus, Hathorn No PES, emp., unphys. values 2002Reduced Dimensionality + Sudden Approx. Charlo, ClarySmall, mostly in wrong direction Second Round of Theoretical Research on Anomalous Isotope Effect 2003Quantum Reactive Scattering Babikov, Walker, Kendrick, Pack J = 0 only 2004Classical Trajectories + adhoc ZPE Schinke and co. Empirical 2005Inelastic Scattering + Sudden Approx. Xie, Bowmanb = 0 only, two orientations 2009Resonance Lifetimes (J > 0) + Strong Collision Approx. Grebenshikov, Schinke Symmetric Top Approx.

9. Statistical Theory of Anomalous Isotope Effect (Marcus) - Densities of states are computed from models (free rotor; hindered rotor). - Simple models for deactivation are assumed (strong collision; step-ladder; exponential).  =1  =1.18  E =210 cm -1 Marcus and co., JCP 117, 1536 (2002). Findings: - Effects of the Y a,b cancel in the “scrambled” conditions and have no effect on enrichments. - The   -effect is essential for enrichments. RRKM Y a,b – partitioning factor ( due to  ZPE );  – symmetry factor ( in either  or  ). - Parameters  E and  are tuned to fit expe- rimental results for 16+1818 and 18+1616. O 3 Na†Na† Nb†Nb† 16+1818 1618+18

10. The First Quantum Dynamics Treatment (Clary) ● Vibrational motion of O 3 is described by the wave function: ● A reduced dimensionality approximation is employed in which the bending angle in Jackoby coordinates is fixed: This wave function is represented numerically by a 2D-grid of points (140 x 140) O 3 bound states resonances O 3 * ~ 90 bound states ~ 60 resonances Charlo and Clary, JCP 117, 1660 (2002). ● Bound states and scattering resonances (metastable states) are computed for J = 0 by solving the TISE using the stabilization method:

11. The collision of O 3 + M is treated by introducing spherical polar coordinates for M: The scattering of M is described by a multi- dimensional wave function in the form: The Coupled-Channel Approach Sudden collision approximation is used (O 3 cant rotate during the collision): - dependence is parametric (essentially 1D). Separate calculations are performed for a large number of fixed values of angles (10 x 24). Final results are obtained by averaging over the angles. Next approximation neglects the ro-vibrational couplings for O 3 states (IOS). same as in O 3

12. Result of the calculations is a scattering matrix for state-to-state transitions: ~ 100 open, ~ 20 closed channels. Solved on a grid of points in R M (~150): Substitution into the TISE leads to a system of coupled-channel equations: The Coupled-Channel Approach - potential coupling matrix - wave vector in channel v Done for ~ 30 total energies E, separately for ℓ = 0 to 150:

13. The First Quantum Results (Clary) ● Results of this very nice work were not particularly encouraging: only one reaction showed large isotope effect in the right direction. In all other cases the isotope effect was either small, or in wrong direction. ● No any correlation was observed. No clear mechanism proposed. ● Due to a number of approximations used it was somewhat hard to figure out the problem. Note: This method was successful in the treatment of HCN + Ar energy transfer. What possibly could be a problem in O 3 case? My guess is: Unlucky combination of the PES features + reduced dimensionality implemented using Jackobi coordinates.

14. Role of Quantum Scattering Resonances (Babikov) ● Focus on energies and lifetimes of O 3 * states. ● Vibrational wave functions are full dimensional (3D): - the adiabatically-adjusting principal-axes hyper-spherical coordinates (APH) are employed. ● Accurate ab initio global PES was used. ● Only the J = 0 case was considered. ● Coupled-Channel treatment of O + O 2 scattering is employed. O + O 2  O 3 * O 3 * + M  O 3 + M

15. Babikov et al, J. Chem. Phys. 118, 6298 (2003). j =1 j =3 j =5 j =7 j =1 j =0 j =2 j =3 j =4 j =5 18 O 16 O 18 O  ZPE O3O3 PES ZPE O 2 O2O2 0. Threshold E O The Lifetime Spectrum

16. j =1 j =3 j =5 j =7 j =1 j =0 j =2 j =3 j =4 j =5 18 O 16 O 18 O  ZPE Energy (eV) Lifetime (ps) 16 O 16 O 18 O j =0 j =1 j =5 j =6 j =3 j =1 j =2 j =3 j =4 j =5 16 O 18 O 16 O  ZPE

17. Babikov et al CPL 372, 686 (2003). 16+1818 PES 1618+18 161818  ZPE Stable O 3 ZPE 1818 ZPE 1618 Mechanism of  ZPE Isotope Effect : 16 O 18 O 18 O

18. 16+1818  ZPE Stable O 3 Metastable O 3 * PES ZPE 1818 ZPE 1618 Babikov et al CPL 372, 686 (2003). 1618+18 161818 Rate: 1.50 Rate: 0.92 Mechanism of  ZPE Isotope Effect : 16 O 18 O 18 O

19. Babikov et al CPL 372, 686 (2003). “Background” Rate: 1.45 Rate: 0.92 16+1618  ZPE Stable O 3 Metastable O 3 * PES ZPE 1618 ZPE 1616 1616+18 161618 Quantum effect; (classical trajectories could not reproduce). General effect. Mechanism of  ZPE Isotope Effect : 16 O 16 O 18 O

20. 3) correct order of magnitude. D. Babikov et al, J. Chem. Phys. 119, 2577, 2003 161818 181616 1) source of a very large isotope effect. 2) is always in the right direction. A simple model: Mechanism of  ZPE Isotope Effect Channel-specific rate coefficients: ; ;

21. Introducing  ZPE into Classical Trajectories (Schinke) Although it is impossible to introduce rigorously the quantum ZPE into classical trajectory simulations, it appears feasible to mimic the  ZPE effect using a simple trick: -  ZPE is introduced into the PES ad hoc; - The time classical trajectories for O + O 2 spend in the O 3 * region is determined; Schinke and Fleurat-Lessard, JCP 122, 94317 (2005). Adjustable parameters describe stabilization and are used to fit the experimental data: - symmetry effect (postulated). - energy transfer efficiency (model), - stabilizing collision frequency ( ~ P ), Smooth dependence, no resonances. - Stabilization probability is defined as:

22. 3D Coupled-Channel Treatment (Bowman) ● Same approach as Clary, but O 3 wave functions are full dimensional (3D), as the PES. A basis set of 100 Legendre polynomials is used for the bending motion. ● Calculations were carried out only for ℓ = 0 (zero impact parameter), single value of collision energy, and three orientations of O 3 * (head, tail, perpendicular). ● The focus was on the  ZPE effect and the role of van der Waals states. Xie and Bowman, Chem. Phys. Lett. 412, 131 (2005). Results: - Confirmed importance of the  ZPE range; - Proposed that the vdW states are important.

23. J > 0 Calculations of the  ZPE Effect (Grebenshchikov) ● Centrifugally Sudden Approximation for rotation and the symmetric top rotor model are used. ● PES is simplified by removing the vdW part, leaving only the covalent well. ● Narrow resonances (  ≤ 1 cm -1 ) are determined for J ≤ 40, K ≤ 10. ● First order perturbation theory is used to determine the branching ratios for two channels. ● Stabilization is not treated, simple model is used (strong collision assumption). Grebenshchikov and Shinke, JCP 131, 181103 (2009). The bottom-line: Several different authors/methods show importance of the  ZPE range. Although not yet modeled with full rigor, this effect appears to be fairly well understood at this point.

24. 2010O + O 2 inelastic scatteringH. Guoexc. agreement with K. Boering New Round of Theoretical and Experimental Research 2010Energy transfer in Ar + O 3 (rotational sudden CC) Ivanov and Schinke no symmetry effect found! YearMethodAuthorComments 1981Experimental DiscoveryMauersberger Stratosphere 2011Energy transfer in Ar + O 3 (mixed quantum-classical) Ivanov and Babikov in progress Focus on: - Explaining very detailed experimental results on O + O 2 scattering studied in the molecular beam conditions; - Finding origin of the symmetry effect (  -factor).

25. Search for Origin of the Symmetry Effect (Schinke) Red – 686 Black – 866 Ivanov and Schinke, Mol. Phys. 108, 259 (2010). ● Sudden Collision Approximation for energy transfer in the Ar + O 3 collisions; IOSA and the Coupled-Channel formalism. ● Similar to Clary/Bowman, but only the bound states (below dissociation threshold) are taken into consideration. Q.: Inaccurate near threshold? ● The wave functions and the PES are full dimensional, but the PES is simplified by removing the vdW part, leaving only the covalent well. Results: Expected symmetry effect was not observed... Agreement with classical trajectory simulations was surprisingly good (small effect in wrong direction).

26. Quantum Symmetry Effect in a Model System (Pack) This simple problem allows to carry out very clean VRIOSA calculations: - masses are slightly modified in order to have exactly the same reduced masses, same energies and lifetimes of all states. - there is no  ZPE here, any difference seen is due to symmetry in the Ne 2 * + M collision. - number of states is small and easy to treat. 16 Ne + 18 Ne  1618 Ne 2 * (+ M)  1618 Ne 2 17 Ne + 17 Ne  1717 Ne 2 * (+ M)  1717 Ne 2 Pack and Walker, JCP 121, 806 (2011). State-to-state ( v, j ) rate coefficients in 1618: Results: - weak (strict) selection rules for state to state transitions in 1618 (1717). Quantum effect, classical trajectories would not reproduce. - this opens additional pathways for the energy transfer and increases the recombination rates. 11% isotope effect !

27. Scaling Problems in the Quantum Dynamics Calculations Scaling with J - Size of the Hamiltonian matrix scales linearly with J : 1296 x 1296 for J=0 41248 x 41248 for J=31 - Cost of calculations scales as ~ J 3, J 2. Kendrick, JCP 114, 8796 (2001). - exponential scaling problem. B. Poirier and co., J. Chem. Phys. 124, 144107 (2006). Scaling with Number of Atoms - Number of vibrational degrees of freedom is 3N–6. - Most QM methods use Direct Product Basis sets (DPB) to express the wave function. As result:

28. Parallelization Problem ( Poirier ) It is found that when the quantum dynamics calculations are parallelized using standard math libraries the efficiency significantly drops after p ~10 or so (communication). Chen and Poirier, J. Comp. Phys. 219 (2006) 185. B. Poirier has shown that using the sparsity pattern of the Hamiltonian matrix, a speedup close to linear and efficiency close to one can be achieved with large number of processors for large systems. ~ 2 x 10 7 - Block Jacoby diagonalization; - Domain decomposition; - Data distribution; - Load balancing.

29. The bending angle is “relaxed” to convert 3D PES into a 2D PES, V(r 1,r 2 ). The overall topology of the surface is preserved: D e, , “reef”, vdW wells, channels. Two channels allow studying the  ZPE effect. The effect of the bending enters through the bending energy correction and the partition function. Adiabatic Bending Model (Babikov)  16 O 16 O + 18 O 16 O + 16 O 18 O Approach based on fast vs. slow degrees of freedom (Born-Oppenheimer like): Ivanov and Babikov, JCP 134, 174308 (2011).

30. Mixed Quantum-Classical Theory of Energy Transfer (Babikov) Ivanov and Babikov, JCP 134, 144107 (2011). The internal vibrational motion is treated with QM using the TDSE (wave packet): resonances,  ZPE and permutation symmetry. The collisional motion O 3 * + M and rotation of O 3 * are treated using classical trajectories: computational advantage. Energy is exchanged between translation, rotation, vibration. Total energy is conserved. 16 O 18 O 16 O J=19, K a =4, K b =12 b(a0)b(a0) Classical degrees of freedom allow intrinsic massive parallelization.

31. Can Quantum Isotope Effects Contribute to S-MIF ? Several gas-phase reactions involve S-atoms and may exhibit the  ZPE and symmetry isotope effects: Gao and Marcus, JCP 127, 244316 (2007). S + S 2 (+ M) → S 3 S m + S n (+ M) → S n+m SO + O (+ M) → SO 2 SO 2 + O (+ M) → SO 3 SO + H (+ M) → HSO SO 2 + OH (+ M) → HSO 3 CI Recombination by ET: Farquhar et al., J. Geophys. Res. 106, 32829 (2001). Pavlov and Kasting, Astrobiology 2, 27 (2002). Several other: S + SH → S 2 + H S + OH → SO + H HS + O → H + SO S 2 + O → S + SO S + O 2 → SO + O

32. Relative Rates (Exp.) (Mauersberger and co., PCCP 3, 4718, 2001)

33. Although not really a one-day job, the classical trajectory simulations can be carried out for variety of chemical reactions relatively easily. Size of the molecule - the number of degrees of freedom, is not really a problem. (Well, given the potential energy surface…) Can Dynamics Methods be Applied to S-MIF ? Ivanov and Schinke, JCP 126, 54304 (2007). Note: It is relatively straightforward to set up such calculations for SO 2 Example: Isotope effect in the O + NO (+ M) → NO 2 recombination. Classical + ZPE method of Schinke was applied and predicted the isotope effect (larger than that in ozone):

34. Construction of the Potential Energy Surfaces Before the nuclear dynamics is studied, the electronic structure problem is solved for many nuclear configurations (independently). Dependence of electronic energy on nuclear configurations gives the potential energy surface. Motion of the nuclei on this surface (dynamics) is studied next. Two major methods for building a continuous surface from the descrete ab initio data points: - Spline (highly accurate, but practical only for small molecules); - Analytic fit (the only way to go in the case of larger polyatomics). Permutational invariance is important in the context of the isotope effects, which affects the choice of - Coordinates; - Functional form of the fitting function.

35. Ground State PES of Ozone Ab initio electronic structure: MOLPRO: icMRCI+Q/cc-pVQZ, CASSCF(12,9). Spline on a 3D grid: 11 x 28 x 20 = 6160 points (Schinke and co., JCP 116, 9749, 2002) Spectroscopically accurate at low energies; wrong behavior in the barrier region. Dissociation energy: D VQZ = 1.027 eV, D EXP = 1.132 eV. O O O 117° 2.4 a 0 “ Reef ” Along the Minimum Energy Path Correction is smooth in 3D; Only upper part of PES; Corrects barrier and D exp. Van-der-Waals tail. (Babikov and co., JCP 118, 6298, 2003)

36. E = – 1.0 eV E = – 0.9 eV E = – 0.8 eV E = – 0.7 eV E = – 0.6 eV E = – 0.5 eV E = – 0.4 eV E = – 0.3 eV E = – 0.2 eV E = – 0.1 eV E = – 0.03 eV E = – 0.02 eV E = – 0.012 eV E = 0 E = – 0.027 eV   E =  ZPE O O O O O O O O O 1. 2. 3. O O O O O O O O O I. II. III. O O O i. ii. O O O ( ,  ) Euler Shape Size Shadow 1D Slice along the MEP: 3D surface: 3D surface

37. Potential Energy Surface of S 3 There is no global potential energy surface available for S 3 at this time. Exploratory work showed many similarities to O 3 : Francisco and co., JCP 123, 54302 (2005). Francisco and co., JCP 125, 84314 (2006). - Isoelectronic, structural similarities; - Two isomers, cyclic-S 3 is at much lower energies, 4.39 kcal/mol. - Calculations at MRCI+Q/CBS are needed to reproduce spec. const. - Covalent well is much deeper, 2.7 eV, vibrational frequencies are smaller (translates into density of states and number of coupled channels). Isotopic shifts predicted for 32 S 3 / 34 S 3 mixture: Analog of the Hartley band in ozone: ~272 nm Their hypothesis: S + S 2 → * S 3 * S 3 → S 2 + * S( 1 D) +hv +M * S( 1 D) → OC * S, * SO 2 +R

38. Potential Energy Surfaces of SO 2 - accurate empirical PES for the ground X 1 A 1 state. H. Guo, Chem. Phys. Lett., 329, 503 (2000). Ab initio PES for the C 1 B 2 state: ~ ~

39. Photo-absorption Spectra of SO 2 Isotopomers (Gua) H. Guo, Chem. Phys. Lett. 439, 280 (2007). bending mode progressions Calculated absorption spectra: - Intensities are quite similar among the isotopomers. - Frequency shifts are regular. Vibrational states of the excited PES (adiabatic calculations) … … … ……

40. Conclusions Very neat quantum mechanical effects lead to MIF in O 3 : -  ZPE effect; - symmetry effect; - scattering resonances. Challenges for theory and experiment on ozone: - spectroscopically accurate PES near O + O 2 threshold; - collisional stabilization of O 3 * and the symmetry effect; - dynamics and spectroscopy near threshold. Work done on the S-containing species: - Predictions of statistical theory for SO 2 ; - PESs of SO 2 ; preliminary work on S 3 ; - Photo-absorption spectra of SO 2. In the near future: - Accurate PESs for S 3 ( MRCI-CBS level ); - Classical trajectory studies for S 3 and SO 2 ( aka Schinke ); - Energy transfer in the S 3 + M collisions ( mixed Q-C ). Acknowledgments: NSF Atmospheric Chemistry Program ($$$)

41.  =0.  =117 deg.  =180 deg.  ~ 80 deg.