Chris Greene Department of Physics and JILA University of Colorado at Boulder

1. How H3+ and its isotopomers recombine efficiently at low energies Chris Greene Department of Physics and JILA University of Colorado at Boulder Main Collaborator: Viatcheslav Kokoouline(U. Central Florida) also with assistance from: Brett Esry support: DOE, NERSC, and NSF The dissociative recombination process:

2. Outline of this talk • Overview of dissociative recombination theory, direct versus indirect processes • Theoretical techniques for the description of molecular Rydberg states • Incorporation of the Jahn-Teller effect and polyatomic dissociation in its full 3D dimensionality • Comparisons between theory and experiment • Remaining problems with the theory to address and overcome.

3. A question of general chemical importance: How does electronic energy convert into bond-breaking energy? …. …. Bates’ 1950 article points out that DR can explain why ionized molecular gases can neutralize so rapidly, even in 10-13 sec or faster.

4. And…. 43 years later… still at the helm: D. R. Bates (1993 “Enigma of H3+ Dissociative Recombination”): “It is concluded that the evidence that the recombination coefficient at 300K is around 1.5 x 10-7 cm3/s is overwhelming. However, such a high rate coefficient has appeared irreconcilable with theory, there being no crossing of potential energy curves favourable to DR at low temperature.”

5. The “direct” DR theory:

6. Systematic inclusion of Rydberg state physics through multichannel quantum defect theory DIRECT INDIRECT via Rydberg states

7. An early calculation of DR for H2+, compared with experiment (Giusti, Derkits, Bardsley 1983): Another foray, slightly later, by Nakashima, Takagi, and Nakamura:

8. The beginnings of understanding electron-ion resonances Prehistory of general resonance physics theory: O. K. Rice, 1933 JCP; Fano, 1935 Nuovo Cimento, 1961 Phys. Rev.; Breit & Wigner, 1936 Phys. Rev.; Blatt & Weisskopf textbook, 1952; Feshbach, 1958,1962, Ann. Phys.; Systematic method for treating Coulomb field aspects of the electron-ion interaction: (Otherwise known as MQDT – multichannel quantum defect theory) Ham, 1955 Solid State Physics; Seaton, 1958 M. Not. R. Astron. Soc.; Seaton, 1983 Rep. Prog. Phys. Fano, Lee, Lu, Johnson, Cheng, 1970s and 1980s PRAs, PRLs. Extensions to molecular physics: Fano, 1970 Phys. Rev. A; Jungen and Atabek, 1977 JCP. Jungen & Dill, 1980 JCP; Including dissociation in molecular MQDT: Lee 1977 PRA; Giusti-Suzor, 1980 J Phys. B; Jungen, 1984 PRL Triatomic Rydberg states: Fano and Lu, 1984 Can. J. Phys; Bordas & Helm, Jungen & Child, JCP; Stephens & Greene, early 1990s;

9. The characteristic Fano resonance lineshape U. Fano 1935, and 1961. H. Feshbach, 1958, 1962 Lorentzian or “Breit-Wigner” limit Two typical resonance plots of an observable versus the energy rescaled by the resonance width.

10. The picture of resonance physics using multichannel quantum defect theory (MQDT) Seaton, Fano, Jungen …; View bound or quasi-bound states as “scattering at E < 0” ionic rovibrational channel thresholds Ev’ Energy(a.u.) closed channels c (E<Ec ) total energy E open channels o (E>Eo ) multichannel quantum defect theory “channel elimination” formula – imposes exponential decay

11. Effects of complicated level perturbations and interlopers, are now fully understood, as in this example of strontium ground state photoionization: Brown, Longmire, Ginter, JOSA B 1983, expt. Aymar, 1987 JPB, theory, using R-matrix theory, multichannel quantum defect theory, and a frame transformation

12. Examples of molecular photofragmentation, for diatomics …An important 1980 JCP article

13. Ch. Jungen and coworkers developed some clever ways to include predissociation on the same footing as preionization of molecular Rydberg state resonances: Recent variations work even better:

14. The “high energy” resonant recombination process is direct, with a strong resonance at 9 eV, accounted for nicely by the Orel-Kulander theoretical treatment. See also A. Larson’s talk Wednesday morning at UCL on the ion-pair formation channel mediated by this resonance.

15. But we’re interested in far cooler temperatures, in the sub-1 eV range. What do experiments say about H3+ dissociative recombination at low-T? • The situation was the following, around the year 2000: • STORAGE RING EXPERIMENTS (@ 300K): • Larsson et al.: 1.15 x 10-7 cm3/s • Mitchell et al.: 1.2 x 10-7 cm3/s • AFTERGLOW EXPERIMENTS (300K): • Gougousi, Johnsen,& Golde, 1995: 1 x 10-8 cm3/s • Laube’, Le Padellec, Rebrion-Rowe, Mitchell, • and Rowe, 1998: 7.8 x 10-8 cm3/s (+/- 2.3) • Smith & Spanel, 1993 1-2 x 10-8 cm3/s • Plasil, Glosik et al. 2002, < 3 x 10-9 cm3/s D. R. Bates (1993 “Enigma of H3+ Dissociative Recombination”): “It is concluded that the evidence that the recombination coefficient at 300K is around 1.5 x 10-7 cm3/s is overwhelming. However, such a high rate coefficient has appeared irreconcilable with theory, there being no crossing of potential energy curves favourable to DR at low temperature.”

16. An important contribution in 2000, by Schneider, Orel, and Suzor-Weiner! Inclusion of Rydberg “indirect pathways, reduces the disagreement to only 3 orders of magnitude! 5 orders of magnitude discrepancy between the “direct pathway” and experiment!

17. In the face of these persistent discrepancies between theory and experiment, and in the face of disagreements between storage ring experiments and afterglow experiments, everyone went back to the drawing board … Experimental improvements to the storage ring experiments: colder ion sources, better energy resolution, especially at CRYRING and TSR Theoretical improvements: Consider Jahn-Teller coupling in the electron-molecule collision, indirect Rydberg pathways, and the full dimensionality of nuclear vibrational motion

18. Our proposed (2001 Nature) mechanism for H3+ dissociative recombination: Jahn-Teller-mediated recombination via Rydberg pathways

19. Two degenerate in-plane p-orbitals are coupled by Jahn-Teller symmetry-distortion physics: Conical intersections, where the non-Born-Oppenheimer couplings blow up real good. Potential Surfaces for dissociative H3 2p,3p states, from M. Jungen

20. Qualitative picture: conversion of the conical intersection problem in 3D to coupled hyperradial potential curves in a single coordinate, the hyperradius, R. H+H+H H3+ H2+H These hyperradial adiabats are derived by solving the fixed-R Schroedinger equation on the dissociative 2p pi surfaces of H3, from Truhlar et al.

21. Why don’t we just do coupled nuclear dynamics on the lower two (H+H2 and H+H+H) dissociative 2p pi surfaces? Because we also have to account for all these Rydberg resonances at low incident electron energies. Here are the np Rydberg series arising from vibrations only.

22. THE BIG PICTURE Hyperspherical representation of the relevant pathways for dissociative recombination of H3+. Notice that in this representation, the DR pathways DO OCCUR AS CURVE CROSSINGS!

23. Clamped-hyperradius electron-H3+ scattering resonances. Shown are the ionic potential curves, and at several hyperradii, the electron scattering time delay as a function of energy.

24. Comparison of the “simplified theoretical treatment” with low resolution experiment Kokoouline, Greene & Esry, Nature 2001 Taken from Kokoouline & Greene, presented in Mosbach: J. Phys.: Conf. Series 4, 74-82 (2005).

25. Next for our “advanced treatment”, a more ambitious approach. Our goal here has been an attempt to: • Perform the first polyatomic dissociative recombination calculation ever that includes ALL 9 degrees of freedom quantum mechanically. (Hopefully resolve the orders-of-magnitude discrepancy between theory and experiment that has existed for this H3+ system for decades.) • Obtain spectroscopic accuracy that can also be compared resonance-by-resonance with H. Helm’s photoionization measurements of metastable H3. (i.e. use the same wavefunctions, but apply them to an observable different from DR.) • Include enough physics to predict the position of most Rydberg state resonances to within 3 meV = 40K, since the astrophysicists want to know this DR rate at T=40K.

26. Scope of the 9D 4-body problem we are faced with:

27. Test of the Staib-Domcke parameterization of the Jahn-Teller fixed-nuclei K-matrix, fitted to ‘undiagonalized’ ab initio calculations of M. Jungen. Figures are taken from Mistrik, Reichle, Muller, Helm, M. Jungen, and J. Stephens, 2000 Phys. Rev. A. Diabatic quantum defect model fitted to the ab initio surfaces, compared with the raw surface values at various geometries

28. Fixed-R S-matrix Approximated as E-independent over 0-2 eV incident energies Smooth MQDT S-matrix for interactions between the electron and the vibrational motion. Note that it is a projection akin to the infinite-order sudden approximation, but we include closed channels in order to get the Rydberg resonance physics. Then add the rotational frame transformation (L-uncoupling physics), and impose exponential decay in the closed ionization channels via MQDT channel elimination:

29. Adiabatic hyperspherical method: Solve for the hyperspherical adiabatic functions using a potential surface for the 3 nuclei in H3+, plotted below for a fixed hyperradius.

30. Adiabatic hyperspherical potential curves for H3+ vibrational motion: Nicely parallel near the equilibrium geometry => Adiabatic approximation should work well! H3+ ab initio surface used from

31. How do we treat molecular Rydberg states while including processes such as photoionization, autoionization, or photodissociation, which involve departures from the Born-Oppenheimer approximation? Answer: Multichannel quantum defect theory (Seaton), combined with a rovibrational frame transformation (Jungen, Fano, Dill, Raoult, Chang, Chase, Arthurs&Dalgarno…) is the only way at present, for many problems, to cope with the immense number of competing channels, and fragmentations of a qualitatively different nature.

32. Concept of an electron-diatomic molecule collision, viewed as a vibrational frame transformation Key Tools in Understanding Rydberg Molecules: Seaton’s multichannel quantum defect theory The Fano-Dill-Jungen rovibrational frame transformation theory  short-range electron-molecule scattering matrix is diagonal in the “quantum number” R.

33. A more complete solution using multichannel quantum defect theory Idea of Fano’s frame transformation: Consider the physical meaning of a scattering matrix in the representation where it is diagonal: => If we can identify in advance the representation in which the S-matrix is diagonal, then we just need to find the eigenphaseshifts and the unitary transformation matrix connecting the eigenchannels to the fragmentation channels . Born-Oppenheimer adiabaticity in electron-diatomic scattering: => the “quantum number” that is conserved during the short-range electron-molecule scattering is R, which gives an S-matrix: An open channel version of this concept was pioneered by Chase 1956, and by Arthurs & Dalgarno 1960.

34. In these states | i > are buried many symmetry considerations, and the full dependence of the wavefunction on vibrational and rotational coordinates, the angular and spin wavefunction of the electron(s), and the nuclear spin wavefunctions. The details are in:

35. A problem: The S-matrix just discussed has only ionization channel indices. How can we represent the dissociation channels? (One way – R-matrix idea of Jungen and Ross.) Our solution: Siegert state methods adapted from Tolstikhin, Ostrovsky, and Nakamura, Phys. Rev. A 58, 2077 (1998).

36. Modified form of the rovibrational frame transformation using a Siegert state vibrational basis to account for the possibility of dissociation.

37. Experiment (red): McCall, Huneycutt, Saykally, Djuric, Dunn, Semaniak, Novotny, Al-Khalili, Ehlerding, Hellberg, Kalhori, Neau, Thomas, Paal, Oesterdahl, Larsson Theory (green), with corrected convolution over , [thanks to Andreas Wolf pointing out its importance]: Kokoouline and Greene (2004) unpublished (no toroidal correction applied yet) This discrepancy is not yet understood This discrepancy is understood – the higher experimental DR rate here is from the toroidal correction

38. Resonance modulations are overestimated by theory because the convolution over delta E(parallel) was not performed (yet). Blowup of the comparison between theory and experiment to better test the crucial region from about 0.05 eV up to 1 eV. Experiment: Sharp drop of DR cross section followed by plateau region 0.4 eV – 0.7 eV shows good agreement between theory and experiment.

39. Origin of the toroidal correction effect

41. Comparison of the ortho H3 photoionization spectrum with the new experimental DR spectrum DR Partial DR, ortho

43. Another prediction of the rotational frame transformation: Average rotational excitation probabilities (squared scattering matrix elements) per p-wave collision, for a low-energy incident electron. Note that many of these are comparable to the unitarity limit (unity).

44. Thermally-averaged rates suggest a possible difference at low temperatures between the destruction rates of ortho and para-H3+.

45. D3+ dissociative recombination calculation, compared with two experiments at different resolutions. Overall, H3+ recombines at a rate about 3 times higher than D3+.

46. Illustration of the stronger nonadiabatic hyperradial coupling in H2D+ compared to H3+. H2D+ H3+ hyperspherical adiabats

47. Vibrational energy levels for H2D+, with and without nonadiabatic hyperradial couplings included H3+ levels

48. Storage ring experiment Calculation in adiabatic approx. Best DR calculation Comparison with results from the storage ring experiments1. The curves represent the DR rate as a function of electronic energy. Experimental conditions are assumed, when theoretical curves were calculated. [1] M. Larsson et al. Phys. Rev. Lett. 79, 395 (1997). Comparison with results from the storage ring experiments1. The curves represent the DR rate as a function of electronic energy. Experimental conditions are assumed, when theoretical curves were calculated. [1] M. Larsson et al. Phys. Rev. Lett. 79, 395 (1997). Comparison with results from the storage ring experiments1. The curves represent the DR rate as a function of electronic energy. Experimental conditions are assumed, when theoretical curves were calculated. [1] M. Larsson et al. Phys. Rev. Lett. 79, 395 (1997).

49. D2H+ dissociative recombination rate versus parallel component of energy This factor of 3-5 discrepancy is not yet understood

50. Comparisons shown on a linear-linear scale for sharper assessment of the extent of disagreement between theory and experiment.