Geometric Phase Effectsin Reaction Dynamics

Department of ChemistryUniversity of Cambridge, UK

Stuart C. Althorpe

Quantum Reaction Dynamics

A

BC

A

BC

ihd(q,Q)

dt ˆ H (q,Q)

ˆ H E

ˆ h ˆ T q U(Q,q)

ˆ H ˆ T Q ˆ h

(q,Q) n (Q)n (q;Q)n

ˆ h n (q;Q) Vn (Q)n (q;Q)

[ ˆ T Q V (Q)](Q) E(Q)

‘clamped nucleus’electronic wave function

Born-Oppenheimer Approximation

A

BC

A

BC

Tnm n (Q) ˆ T Q m (Q)

B.-O.: assume v. small

Potential energyNuclear dynamics S.E.

exact:

V (Q)

Reactive Scattering

A

BC

A

BC

resonances

rearrangement

3 or 4 atom reactions

H + H2O OH + H2

H + H2 H2 + H

H + HX H2 + X

nme ikR Snme ikR

S nme ik R

e i ˆ H t /h [e iKt / 2Nhe iVt / Nhe iKt / 2Nh]N

scattering b.c.

V (Q)AB + C

A + BC

propagator

R

R

(q,Q) 0(Q)0(q;Q) 1(Q)1(q;Q)

(q,Q) 0(Q)0(q;Q)

ˆ T 00 V0ˆ T 01

ˆ T 10ˆ T 11 V1

0

1

E

0

1

(Group) Born-Oppenheimer Approximation

ˆ T 01 1(Q) ˆ T Q 0(Q) not small

conicalintersection

derivative coupling terms

V0(Q)

V1(Q)

‘Non-crossing rule’

XV0

V1

Conical intersections

‘Non-crossing rule’

V0

V1

‘N − 2 rule’ N = 2N = 3

N = 1

Herzberg & Longuet-Higgins (1963)

(q,Q) 0(Q)0(q;Q) 1(Q)1(q;Q)

Geometric (Berry) Phase

n ( 2N ) ( 1)N n ()

n ()

— 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

n = 0

n = −1

Schulman, Phys Rev 1969; Phys Rev D 1971; DeWitt, Phys Rev D 1971

Winding number of Feynman paths

K(x,x0,t) = Ke(x,x0,t) + Ko(x,x0,t)

Ψ(x,t) = Ψe(x,t) + Ψo(x,t)

Ψ(x,t) = dx0 K(x,x0,t) Ψ(x0,0)

K(x,x0,t) = Σ eiS/ħ

∫path

n = 0

n = -1

K(x,x0,t) = Ke(x,x0,t) + Ko(x,x0,t)

Ψ(x,t) = Ψe(x,t) + Ψo(x,t)

−

−

Ψe(x,t) Ψo(x,t)

repeat calculationwith and without cut-line

Bound-state BC

Scattering BC

() exp(im)

() exp[i(m 1/2)]

cut-line

H + H2 HH + H

+

H + H2 HH + H

HAHC + HB

+

‡ ‡

‡HAHB + HC

Ψo Ψe

HA + HBHC

+ HBHCHA

H + H2 HH + H

Ψe

Ψo

Internal coordinates Scattering angles

dd

(,E)differential cross section

∞

(R,r,,,,)

+ HBHCHA

H + H2 HH + H

Ψe

Ψo

Internal coordinates Scattering angles

(R,r,,,,)

Scattering experimentsZare (Stanford), Yang (Dalian)

J.C. Juanes-Marcos, SCA, E. Wrede, Science 2005

0021

F. Bouakline, S.C. Althorpe and D. Peláez Ruiz, JCP  (2008).

2.3 eV

3.0 eV

4.0 eV

4.3 eV

DC

S (

Ǻ2 S

r-1)

+

‡ ‡

‡

Ψe

Ψo

High collision energy

Conical intersections

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

+

Discontinuous paths?Simply connected?

on two coupled surfaces?

ΨeΨo

+

very small

ΨoΨe

Ψ = Ψe + Ψo

Ψ = Ψe − Ψo~

Geometric phase

+

Discontinuous paths?

on two coupled surfaces?

ΨeΨo

✓

K(s,x;s0,x0|t) = ∑….∑∑S1S2SN

K(s,sN….s2,s1,s0;x,x0|t)

Time-ordered product

S=0

S=1+

S=1

S=0

SCA, Stecher, Bouakline, J Chem Phys 2008

=x0

x

P. Pechukas, Phys Rev 1969

n = 0

+

on two coupled surfaces?

ΨeΨo

✓ ✓

Ψo Ψe

on two coupled surfaces

ΨeΨo

+

Ψo Ψe

+

Ψo Ψe

P0/P1

1.25

1.93

S=0

S=1

Pyrrole

1B1(πσ*)-S0 Conical Intersection (surfaces of Vallet et al. JCP 2005)

N

H

Negligible phase effectson population transfer

GP-enhancedrelaxation

Conclusions

GP effects small in reaction dynamics except possibly:

• at low temperatures

• in short-time quantum control experiments

Dr Foudhil Bouakline

Thomas Stecher

Thanks for listening