1. Martin Čížek Charles University, Prague Non-Local Nuclear Dynamics Dedicated to Wolfgang Domcke and Jiří Horáček 1348

2. Studied processes: AB(v) + e -  AB(v’  v) + e - (VE) AB(v) + e -  A + B - (DA) A + B -  AB(v) + e - (AD) AB(v) + e -  (AB) -  A + B -

3. Outline of Theory Review: W. Domcke, Phys. Rep. 208 (1991) 97 http://utf.mff.cuni.cz/~cizek/ Fixed nuclei calculation as a first step. Fano-Feshbach projection to get the electronic basis. Known analytic properties of matrix elements (threshold expansions) used to construct proper model. Nuclear dynamics solved assuming diabaticity of basis.

4. Electronic structure for fixed-R A + + B - A + B Negative ion system (HCl)-Two state Landau-Zener model H + Cl - HCl + e - Main idea behind the theoretical approach (O’Malley 1966): Selection of proper diabatic electronic basis set consisting of anionic discrete state and (modified) electron scattering continuum

5. Extraction of resonance from the continuum Essence of the method: Selection of a square integrable function (discrete state) describing approximately the resonance and solution of scattering problem with additional constraint (orthogonality to the discrete state) It is show that sharp resonance structures are removed from continuum with sensitive choice of discrete state Example: Scattering of particle from spherical delta-shell. Discrete states – bound states in box with the same size as the shell.

6. Discrete state … Continuum … Coupling Diabaticity of the basis: Hamiltonian in the basis: Final diabatic basis set

7. Nonlocal vibrational dynamics in (AB) - state Expansion of wave function Projection Schrödinger equation on basis Formal solution of second line for   (R) into first line The similar procedure for Lippmann-Schwinger equation yields:

8. where Threshold behavior Equation of motion for nuclei

9. Nonlocal resonance model Dynamics is fully determined by knowledge of the functions V 0 (R), V d (R), V d  (R) It is convenient to define: Then it is

10. Summary – our procedure Model parameters V 0 (R), V d (R) and V dε (R) found from Fano-Feshbach or fit for fixed-nuclei Analytic fit made for R and e-dependencies in V dε (R) to be able to perform the transform and efficient potential evaluation Nuclear dynamics is solved for ψ d (R) component Cross sections or other interesting quantities are evaluated

11. Results HCl(v) + e -  HCl(v’) + e - (VE)

12. Results – vibrational excitation in e - + HCl Integral cross section. Theory versus measurement of Rohr, Linder (1975) and Ehrhardt (1989) Differential cross section. Measurement of Schafer and Allan (1991)

13. Results – vibrational excitation in e - + HCl Elastic cross section. Theory -- resonant contribution (top) versus measurement of Allan 2000 (bottom) Vibrational excitation 0->1. Theory (top) versus measurement of Allan 2000 (bottom)

14. VE in e - +H 2

15. Interpretation of boomerang oscillations Dashed line = neutral molecule potential Solid line = negative ion – discrete state potential Circles = ab initio data for molecular anion Boomerang oscillations: interference of direct process and reflection from long range part of anion potential

16. Results HBr(v) + e -  H + Br - (DA)

17. Results – DA to HBr and DBr Comparison with measurement of Sergenton and Allan 2001

18. Results H 2 + e - ↔ H 2 - ↔ H - +H

19. M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571 H+H - → e - + H 2

20. M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571

21. Potentials for J=0 Potential V ad (R) for nonzero J The Origin of the Resonances

22. Cross section AUTODETACHMET Resonant tunneling wave function Energy V ad (R) + J(J+1)/2 μ R 2

23. Elastic cross section for e - + H 2 (J=21, v=2)

24. Γ 0 =2.7×10 -4 eV

25. Elastic cross section for e - + H 2 (J=25, v=1)

26. Γ 0 =2.7×10 -9 eV Γ 1 =1.9×10 -6 eV

27. Table I: Parameters of H 2 - states JE res (relative to DA)τ 21-136 meV2.4 ps 22-105 meV12 ps 23-75 meV0.11 ns 24-47 meV0.9 ns 25-20 meV12 ns 265 meV0.52 μs 2728 meV2 ns

28. Table II: Parameters of D 2 - states JE res (relative to DA)τ 31-118 meV0.13 ns 32-97 meV0.70 ns 33-76 meV6 ns 34-55 meV39 ns 35-35 meV0.51 μs 36-16 meV5.7 μs 372 meV14 μs 3819 meV7.2 μs 3934 meV41 ps