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Geometric Phase Effects in Reaction Dynamics

Geometric Phase Effects in Reaction Dynamics. Stuart C. Althorpe. Department of Chemistry University of Cambridge, UK. Quantum Reaction Dynamics. B. B. C. C. A. A. Born-Oppenheimer Approximation. B. B. C. C. A. A. ‘clamped nucleus’ electronic wave function. exact:.

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Geometric Phase Effects in Reaction Dynamics

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  1. Geometric Phase Effectsin Reaction Dynamics Stuart C. Althorpe Department of Chemistry University of Cambridge, UK

  2. Quantum Reaction Dynamics B B C C A A

  3. Born-Oppenheimer Approximation B B C C A A ‘clamped nucleus’electronic wave function exact: B.-O.: assume v. small Potential energy Nuclear dynamics S.E.

  4. Reactive Scattering rearrangement B B C C A A scattering b.c. resonances A + BC AB + C 3 or 4 atom reactions propagator H + H2 H2 + H H + HX  H2 + X H + H2O  OH + H2

  5. (Group) Born-Oppenheimer Approximation not small conicalintersection derivative coupling terms

  6. Conical intersections ‘Non-crossing rule’ V1 X V0

  7. ‘Non-crossing rule’ ‘N − 2 rule’ N = 3 N = 2 N = 1 V1 V0

  8. Geometric (Berry) Phase Herzberg & Longuet-Higgins (1963) — double-valued BC cut-line Aharanov-Bohm

  9. ∫ Ψ(x,t) =dx0 K(x,x0,t) Ψ(x0,0) K(x,x0,t) = ΣeiS/ħ path K(x,x0,t) = Ke(x,x0,t)+ Ko(x,x0,t) Ψ(x,t)= Ψe(x,t) + Ψo(x,t) n = −1 n = 0 Winding number of Feynman paths Schulman, Phys Rev 1969; Phys Rev D 1971; DeWitt, Phys Rev D 1971

  10. ∫ Ψ(x,t) =dx0 K(x,x0,t) Ψ(x0,0) K(x,x0,t) = ΣeiS/ħ − − path K(x,x0,t) = Ke(x,x0,t)+ Ko(x,x0,t) Ψ(x,t)= Ψe(x,t) + Ψo(x,t) n = -1 n = 0 repeat calculationwith and without cut-line Ψe(x,t) Ψo(x,t)

  11. Bound-state BC Scattering BC cut-line

  12. H + H2 HH + H +

  13. H + H2 HH + H HA + HBHC ‡ ‡ + Ψo Ψe HAHB + HC ‡ HAHC + HB

  14. H + H2 HH + H ∞ Ψe q Ψo HA HBHC + differential cross section Internal coordinates Scattering angles

  15. H + H2 HH + H Ψe Ψo HA HBHC + Scattering experiments Zare (Stanford), Yang (Dalian) Internal coordinates Scattering angles J.C. Juanes-Marcos, SCA, E. Wrede, Science 2005

  16. 0021 High collision energy 2.3 eV 3.0 eV ‡ ‡ Ψe DCS (Ǻ2Sr-1) 4.0 eV + Ψo 4.3 eV ‡ F. Bouakline, S.C. Althorpe and D. Peláez Ruiz, JCP(2008).

  17. Conical intersections Domcke, Yarkony, Köppel (eds)Conical Intersections(World Scientific, New Jersey, 2003).

  18. Ψo Ψe on two coupled surfaces? + Simply connected? Discontinuous paths?

  19. Ψo + Ψe Ψ= Ψe+Ψo ~ very small Ψ= Ψe−Ψo Geometric phase

  20. Ψo Ψe on two coupled surfaces? + ✓ Discontinuous paths?

  21. Time-ordered product P. Pechukas, Phys Rev 1969 = ∑….∑∑ K(s,x;s0,x0|t) K(s,sN….s2,s1,s0;x,x0|t) SN S2 S1 S=1 x0 S=1 = x + S=0 S=0 n = 0 SCA, Stecher, Bouakline, J Chem Phys 2008

  22. Ψo Ψe on two coupled surfaces? + ✓ ✓

  23. Ψo Ψe on two coupled surfaces Ψe Ψo

  24. Ψo Ψe +

  25. Ψo Ψe +

  26. S=1 S=0 P0/P1 1.93 1.25

  27. Pyrrole H N 1B1(πσ*)-S0 Conical Intersection (surfaces of Vallet et al. JCP 2005) Negligible phase effects on population transfer

  28. GP-enhancedrelaxation

  29. Conclusions GP effects small in reaction dynamics except possibly: • at low temperatures • in short-time quantum control experiments

  30. Thanks for listening Dr Foudhil Bouakline Thomas Stecher

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