R-matrix theory and

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R-matrix theory and Electron-molecule
scattering
e-
Jonathan Tennyson Department of Physics and
Astronomy University College London
Outer region
Inner region
UCL, May 2004
Lecture course on open quantum systems
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What is an R-matrix?
Consider coupled channel equation
Use partial wave expansion hi,j(r,q,f) Plm
(q,f) uij(r) Plm associated Legendre functions
where
General definition of an R-matrix
where b is arbitrary, usually choose b0.
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R-matrix propagation
Asymptotic solutions have form
open channels
closed channels
R-matrix is numerically stable
For chemical reactions can start from Fij 0 at
r 0 Light-Walker propagator J. Chem. Phys. 65,
4272 (1976).
Also Baluja, Burke Morgan, Computer Phys.
Comms., 27, 299 (1982) and 31, 419 (1984).
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Wigner-Eisenbud R-matrix theory
Outer region
e-
H H
Inner region
R-matrix boundary
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Consider the inner region
Schrodinger Eq
Finite region introduces extra surface operator
Bloch term
for spherical surface at r x b arbitrary.
Necessary to keep operator Hermitian.
Schrodinger eq. for finite volume becomes
which has formal solution
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Eq. 1
Expand u in terms of basis functions v
Coefficients aijk determined by solving
Inserting this into eq. 1
Eq. 2
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R-matrix on the boundary
Eq. 2 can be re-written using the R-matrix
which gives the form of the R-matrix on a surface
at r x
in atomic units, where Ek is called an R-matrix
pole uik is the amplitude of the channel
functions at r x.
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Why is this an R-matrix?
In its original form Wigner, Eisenbud others
used it to characterise resonances in nuclear
reactions. Introduced as a parameterisation
scheme on surface of sphere where processes
inside the sphere are unknown.
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Resonancesquasibound states in the continuum

• Long-lived metastable state where the scattering
  electron is temporarily captured.
• Characterised by increase in p in eigenphase.
• Decay by autoionisation (radiationless).
• Direct Indirect dissociative recombination
  (DR), and other processes, all go via resonances.
• Have position (Er) and width (G)
• (consequence of the Uncertainty Principle).
• Three distinct types in electron-molecule
  collisions
• Shape, Feshbach nuclear excited.

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Electron molecule collisions
Outer region
e-
H H
Inner region
R-matrix boundary
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Dominant interactions
Inner region
Exchange Correlation
Adapt quantum chemistry codes
High l functions required Integrals over finite
volume Include continuum functions Special
measures for orthogonality CSF generation must be
appropriate
Boundary
Target wavefunction has zero amplitude
Outer region
Adapt electron-atom codes
Long-range multipole polarization potential
Many degenerate channels Long-range (dipole)
coupling
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Inner region Scattering wavefunctions
Yk A Si,j ai,j,k fiN hi,j Sm bm,k fmN1
where
fiN N-electron wavefunction of ith target state
hi,j 1-electron continuum wavefunction
fmN1 (N1)-electron short-range functions L2
ai,j,k and bj,k variationally determined
coefficients
A Antisymmetrizes the wavefunction
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Target Wavefunctions
fiN Si,j ci,jzj
where
fiN N-electron wavefunction of ith target state
zj N-electron configuration state function
(CSF) Usually defined using as CAS-CI
model. Orbitals either generated internally or
from other codes
ci,j variationally determined coefficients
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Continuum basis functions
Use partial wave expansion (rapidly
convergent) hi,j(r,q,f) Plm (q,f) uij(r) Plm
associated Legendre functions

• Diatomic code l any, in practice l lt 8
• u(r) defined numerically using boundary
  condition u(ra) 0
• This choice means Bloch term is zero but
• Needs Buttle Correction..not strictly
  variational
• Schmidt Lagrange orthogonalisation
• Polyatomic code l lt 5
• u(r) expanded as GTOs
• No Buttle correction required..method
  variational
• But must include Bloch term
• Symmetric (Lowden) orthogonalisation

Linear dependence always an issue
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R-matrix wavefunction
Yk A Si,j ai,j,k fiN hi,j Sm bm,k fmN1
only represents the wavefunction within the
R-matrix sphere
ai,j,k and bj,k variationally determined
coefficients by diagonalising inner region
secular matrix. Associated energy (R-matrix
pole) is Ek.
Full, energy-dependent scattering wavefunction
given by
Y(E) Sk Ak(E) Yk
Coefficients Ak determined in outer region (or
not) Needed for photoionisation, bound states,
etc. Numerical stability an issue.
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R-matrix outer region K-, S- and T-matrices
Propagate R-matrix (numerically v. stable)
Asymptotic boundary conditions
Open channels
Closed channels
Defines the K (reaction)-matrix. K is real
symmetric. Diagonalising K ? KD gives the
eigenphase sum
Use eigenphase sum to fit resonances
Eigenphase sum
The K-matrix can be used to define the S
(scattering) and T (transition) matrices.
Both are complex.
, T S - 1
Use T-matrices for total and differential cross
sections
S-matrices for Time-delays MQDT analysis
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UK R-matrix codes www.tampa.phys.ucl.ac.uk/rmat
SCATCI Special electron Molecule
scattering Hamiltonian matrix construction
L.A. Morgan, J. Tennyson and C.J. Gillan,
Computer Phys. Comms., 114, 120 (1999).
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Non-adiabatic configuration space
Electron-molecule coordinate r
H2 e-
H H e-
Electronic R-matrix Boundary a
Internal region
H H-
Double R-matrix method
Internuclear distance R
0
Ain
Aout
Nuclear R-matrix boundaries
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Processes at low impact energies
Elastic scattering AB e AB
e
Electronic excitation AB e
AB e
Vibrational excitation AB(v0) e
AB(v) e
Rotational excitation AB(N) e
AB(N) e
Dissociative attachment / Dissociative
recombination AB e A? B
A B?
Impact dissociation AB e A
B e
All go via (AB-) . Can also look for bound
states
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Electron - LiH scattering 2S eigenphase sums
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Pseudo Resonances

• Unphysical resonances at higher energies
• Present in any calculation with polarisation
  effects
• Occur above lowest state omitted from calculation
• Always a problem above ionisation threshold
• Effects can be removed by averaging
• eg Intermediate Energy R-Matrix (IERM)
  method