Martin Čížek Charles University, Prague Non-Local Nuclear Dynamics Dedicated to Wolfgang Domcke and Jiří Horáček 134 8
Dec 22, 2015
Martin Čížek
Charles University, Prague
Non-Local Nuclear Dynamics
Dedicated to Wolfgang Domcke and Jiří Horáček
1348
Studied processes:
AB(v) + e- AB(v’ v) + e- (VE)
AB(v) + e- A + B- (DA)
A + B- AB(v) + e- (AD)
AB(v) + e- (AB)- A + B-
Outline of Theory
Review: W. Domcke, Phys. Rep. 208 (1991) 97
http://utf.mff.cuni.cz/~cizek/
• Fixed nuclei calculation as a first step.
• Fano-Feshbach projection to get the electronic basis.
• Known analytic properties of matrix elements (threshold expansions) used to construct proper model.
• Nuclear dynamics solved assuming diabaticity of basis.
Electronic structure for fixed-R
A+ + B-
A + B
Negative ion system (HCl)- Two state Landau-Zener model
H + Cl-
HCl + e-
Main idea behind the theoretical approach (O’Malley 1966):
Selection of proper diabatic electronic basis set consisting of anionic discrete state and (modified) electron scattering continuum
Extraction of resonance from the continuum
Essence of the method:
Selection of a square integrable function (discrete state) describing approximately the resonance and solution of scattering problem with additional constraint (orthogonality to the discrete state)
It is show that sharp resonance structures are removed from continuum with sensitive choice of discrete state
Example:
Scattering of particle from spherical delta-shell. Discrete states – bound states in box with the same size as the shell.
Discrete state …
Continuum …
Coupling
Diabaticity of the basis:
Hamiltonian in the basis:
Final diabatic basis set),( rRd
),( rR
)(RVH ddeld
)'())(( 0' RVH el
)(RVH deld
0),( ,0),(
rRR
rRR d
dVVVVTH dddddddN )( *0
0
0)(
0
)(0
0
)(0
0
0)(
*0
0 RV
RV
TRV
RVT
VHd
d
N
dN
Nonlocal vibrational dynamics in (AB)- state
• Expansion of wave function
),()( ),()(),( rRRdrRRrR d
• Projection Schrödinger equation on basis
)( )()( )(
)( )( )( )( 0),()(*
0 RRVRRVTE
RRVdRRVTErRHE
dN
ddN
• Formal solution of second line for (R) into first line
)'(')( )',,( where
0)( )',,(')(
*10 RVRiVTERRVdRREF
RRREFdRRVTE
dNd
dN
• The similar procedure for Lippmann-Schwinger equation yields:
)AB(vefor )(
BAfor e wher,)( -
0
-1
ivd
iKR
dN
iiVEG
eFViTE
wherewhere
Threshold behaviorThreshold behavior
)()(2
)1(
2
122
2
RERVR
JJ
R dd
0)'()'()',,(')( RRRREEfdRR dJv
Jv
v
Jv
Equation of motion for nuclei
)'()()0'(')',,( *''
1 RVRVidRRf dd
)(~)(RVd
210 :scattering dipole
21 :scattering wave-s
23 :scattering wave-p
Nonlocal resonance modelDynamics is fully determined by knowledge of the functions
V0(R), Vd(R), Vd(R)
)'/(),'( '..1
),( |)(|2),( 2
RdpvRRVR d
')(2
)()',,(0
Ri
RRREF VTE N
It is convenient to define:
Then it is)(
2)( R
iR
Summary – our procedure
• Model parameters V0(R), Vd(R) and Vdε(R) found from Fano-Feshbach or fit for fixed-nuclei
• Analytic fit made for R and e-dependencies in Vdε(R) to be able to perform the transform
and efficient potential evaluation
• Nuclear dynamics is solved for ψd(R) component
• Cross sections or other interesting quantities are evaluated
)'()()0'(')',,( *''
1 RVRVidRRf dd
Results – vibrational excitation in e- + HCl
Integral cross section. Theory versus measurement of Rohr, Linder (1975) and Ehrhardt (1989)
Differential cross section.
Measurement of Schafer and Allan (1991)
Results – vibrational excitation in e- + HCl
Elastic cross section.
Theory -- resonant contribution (top) versus measurement of Allan 2000 (bottom)
Vibrational excitation 0->1.
Theory (top) versus measurement
of Allan 2000 (bottom)
Interpretation of boomerang oscillations
• Dashed line = neutral molecule potential
• Solid line = negative ion – discrete state potential
• Circles = ab initio data
for molecular anion
Boomerang oscillations:
interference of direct process
and reflection from long range
part of anion potential
Cross section
AU
TO
DE
TA
CH
ME
T
Resonant tunneling wave function E
ner
gy
)(Rd
Vad(R) + J(J+1)/2μR2
Table I: Parameters of H2- states
J Eres (relative to DA) τ
21 -136 meV 2.4 ps
22 -105 meV 12 ps
23 -75 meV 0.11 ns
24 -47 meV 0.9 ns
25 -20 meV 12 ns
26 5 meV 0.52 μs
27 28 meV 2 ns
Table II: Parameters of D2- states
J Eres(relative to DA) τ
31 -118 meV 0.13 ns
32 -97 meV 0.70 ns
33 -76 meV 6 ns
34 -55 meV 39 ns
35 -35 meV 0.51 μs
36 -16 meV 5.7 μs
37 2 meV 14 μs
38 19 meV 7.2 μs
39 34 meV 41 ps