Martin ČížekCharles 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),(

rR

RrR

R d

dVVVVTH dddddddN )( *0

00)(

0)(0

0)(0

00)(

*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 RRVRRVTERRVdRRVTErRHE

dN

ddN

• Formal solution of second line for (R) into first line

)'(')( )',,( where

0)( )',,(')( *1

0 RVRiVTERRVdRREF

RRREFdRRVTE

dNd

dN

• The similar procedure for Lippmann-Schwinger equation yields:

)AB(vefor )(BAfor

e wher,)( -0

-1

ivd

iKR

dNii

VEGe

FViTE

wherewhere

Threshold behaviorThreshold behavior

)()(2

)1(21

22

2

RERVR

JJR 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

RiRRREF VTE N

It is convenient to define:

Then it is)(

2)( RiR

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 HCl(v) + e- HCl(v’) + e- (VE)

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)

VE in e-+H2

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 rangepart of anion potential

Results HBr(v) + e- H + Br- (DA)

Results – DA to HBr and DBr

Comparison with measurement of Sergenton and Allan 2001

Results

H2 + e- ↔ H2- ↔ H-+H

M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571

H+H- → e- + H2

M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571

Potentials for J=0 Potential Vad(R) for nonzero J

The Origin of the Resonances

Cross section

AU

TOD

ETA

CH

MET

Resonant tunneling wave function En

ergy

)(Rd

Vad(R) + J(J+1)/2μR2

Elastic cross section for e- + H2 (J=21, v=2)

Γ0=2.7×10-4eV

Elastic cross section for e- + H2 (J=25, v=1)

Γ0=2.7×10-9eV

Γ1=1.9×10-6eV

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