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

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

(Group) Born-Oppenheimer Approximation not small conical intersection derivative coupling terms

Conical intersections ‘Non-crossing rule’ V1 X V0

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

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

∫ 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

∫ 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)

Bound-state BC Scattering BC cut-line

H + H2  HH + H +

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

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

+ 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

+ 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).

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

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

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

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

+ 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

Ψo Ψe on two coupled surfaces? + ✓ ✓

Ψo Ψe on two coupled surfaces Ψe Ψo

Ψo Ψe +

Ψo Ψe +

S=1 S=0 P0/P1 1.93 1.25

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

GP-enhanced relaxation

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

Thanks for listening Dr Foudhil Bouakline Thomas Stecher