geometric phase effects in reaction dynamics
DESCRIPTION
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:. - PowerPoint PPT PresentationTRANSCRIPT
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