Transcript Document

R-matrix theory and
Electron-molecule scattering
eJonathan Tennyson
Department of Physics and Astronomy
University College London
Outer region
Inner region
UCL, May 2004
Lecture course on open quantum systems
What is an R-matrix?
Consider coupled channel equation:
where
Use partial wave expansion
hi,j(r,q,f) = Plm (q,f) uij(r)
Plm associated Legendre functions
General definition of an R-matrix:
where b is arbitrary, usually choose b=0.
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).
Wigner-Eisenbud R-matrix theory
Outer region
eH
H
Inner region
R-matrix boundary
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
Eq. 1
Expand u in terms of basis functions v
Coefficients aijk determined by solving
Inserting this into eq. 1
Eq. 2
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.
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.
Resonances:
quasibound 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.
Electron – molecule collisions
Outer region
eH
H
Inner region
R-matrix boundary
Dominant interactions
Inner region
Exchange
Correlation
Boundary
Adapt quantum chemistry codes
High l functions required
Integrals over finite volume
Include continuum functions
Special measures for orthogonality
CSF generation must be appropriate
Target wavefunction has zero amplitude
Outer region Adapt electron-atom codes
Long-range multipole polarization potential
Many degenerate channels
Long-range (dipole) coupling
Inner region: Scattering wavefunctions
Yk = A Si,j ai,j,k fiN hi,j + Sm bm,k fmN+1
where
fiN N-electron wavefunction of ith target state
hi,j 1-electron continuum wavefunction
fmN+1 (N+1)-electron short-range functions ‘L2’
ai,j,k and bj,k variationally determined coefficients
A Antisymmetrizes the wavefunction
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
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 < 8
u(r) defined numerically using boundary condition u’(r=a) = 0
This choice means Bloch term is zero but
Needs Buttle Correction…..not strictly variational
Schmidt & Lagrange orthogonalisation
Linear dependence
• Polyatomic code: l < 5
always an issue
u(r) expanded as GTOs
No Buttle correction required…..method variational
But must include Bloch term
Symmetric (Lowden) orthogonalisation
R-matrix wavefunction
Yk = A Si,j ai,j,k fiN hi,j + Sm bm,k fmN+1
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.
R-matrix outer region:
K-, S- and T-matrices
Asymptotic boundary conditions:
Propagate R-matrix
(numerically v. stable)
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.
S-matrices for
Time-delays &
MQDT analysis
,T=S-1
Use T-matrices for
total and differential
cross sections
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).
Non-adiabatic configuration space
Electron-molecule
coordinate r
H 2 + e-
H + H + e-
Electronic R-matrix
Boundary a
Internal region
Double
R-matrix
method
0
Ain
Aout
Nuclear R-matrix boundaries
H + HInternuclear
distance R
Processes: at low impact energies
Elastic scattering
AB + e
AB + e
Electronic excitation
AB + e
AB* + e
Vibrational excitation
AB(v”=0) + e
AB(v’) + e
Rotational excitation
AB(N”) + e
AB(N’) + e
Dissociative attachment / Dissociative recombination
AB + e
A- + B
A + BImpact dissociation
AB + e
A+B+e
All go via (AB-)** . Can also look for bound states
Electron - LiH scattering:
2S
eigenphase sums
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