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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