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

2. Quantum Reaction DynamicsB B C C A A

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

4. Reactive Scattering B B C C A A rearrangement scattering b.c. A + BC
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
conical intersection
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) = Σ eiS/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) = Σ eiS/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 calculation with 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 Ψo q ∞ HA HBHC Internal coordinates
differential cross section
Internal coordinates
Scattering angles

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

16. + Ψe Ψo High collision energy ‡ ‡ ‡ 0021 2.3 eV 3.0 eV DCS (Ǻ2Sr-1)
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?

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

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

22. + Time-ordered product = ∑….∑∑ K(s,x;s0,x0|t) K(s,sN….s2,s1,s0;x,x0|t)
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

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

24. Ψo Ψe
on two coupled surfaces Ψe Ψo

26. Ψo Ψe +

27. Ψo Ψe +

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

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

30. GP-enhanced relaxation

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

32. Thanks for listening
Dr Foudhil Bouakline
Thomas Stecher