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The R-matrix Method for oriented molecules. e -. Jonathan Tennyson University College London. Jonathan Tennyson Department of Physics and Astronomy University College London. Outer region. Inner region. UCL, May 2004. Montreal Oct 2009. Lecture course on open quantum systems. - PowerPoint PPT Presentation
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Jonathan Tennyson
Department of Physics and Astronomy
University College London
Lecture course on open quantum systems
e-
UCL, May 2004
Inner region
Outer region
The R-matrix Methodfor oriented molecules
Jonathan Tennyson University College London
MontrealOct 2009
What is an R-matrix?
General definition of an R-matrix:
where b is arbitrary, usually choose b=0.
Consider coupled channel equation:
whereUse partial wave expansion hi,j(r,q,f) = Plm (q,f) uij(r)Plm associated Legendre functions
R-matrix propagationAsymptotic solutions have form:
R-matrix is numerically stable
For chemical reactions can start from Fij = 0 at r = 0Light-Walker propagator: J. Chem. Phys. 65, 4272 (1976).
Also: Baluja, Burke & Morgan, Computer Phys. Comms.,27, 299 (1982) and 31, 419 (1984).
open channels
closed channels
H H
e
Inner region
Outer region
R-matrix boundary
Wigner-Eisenbud R-matrix theory
Consider the inner region
Schrodinger Eq:
Finite region introduces extra surface operator:
Bloch term:
Schrodinger eq. for finite volume becomes:
which has formal solution
for spherical surface at r = x; b arbitrary. Necessary to keep operator Hermitian.
Expand u in terms of basis functions v
Coefficients aijk determined by solving
Eq. 1
Inserting this into eq. 1
Eq. 2
R-matrix on the boundaryEq. 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, whereEk 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 itto characterise resonances in nuclear reactions.Introduced as a parameterisation scheme on surface ofsphere 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 in eigenphase.
• Decay by autoionisation (radiationless).
• Direct & Indirect dissociative recombination (DR), and other processes, all go via resonances.
• Have position (Er) and width ()
(consequence of the Uncertainty Principle).
• Three distinct types in electron-molecule collisions:
Shape, Feshbach & nuclear excited.
H H
e-
Inner region
Outer region
R-matrix boundary
Electron – molecule collisions
Dominant interactionsInner region
Exchange Correlation
Adapt quantum chemistry codes
Outer regionLong-range multipole polarization potential
Adapt electron-atom codes
High l functions requiredIntegrals over finite volumeInclude continuum functionsSpecial measures for orthogonalityCSF generation must be appropriate
Many degenerate channelsLong-range (dipole) coupling
Boundary Target wavefunction has zero amplitude
Inner region: Scattering wavefunctions
kA i,jai,j,kiNi,jm bm,km
N+1where
iN N-electron wavefunction of ith target state
i,j1-electron continuum wavefunction
mN+1 (N+1)-electron short-range functions ‘L2’
A Antisymmetrizes the wavefunction
ai,j,kand bj,kvariationally determined coefficients
Target Wavefunctionsi
N = i,j ci,jj
where
iN N-electron wavefunction of ith target state
j N-electron configuration state function (CSF) Usually defined using as CAS-CI model.
Orbitals either generated internally or from other codes
ci,jvariationally determined coefficients
Continuum basis 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
• Polyatomic code: l < 5
u(r) expanded as GTOs
No Buttle correction required…..method variational
But must include Bloch term
Symmetric (Lowden) orthogonalisation
Use partial wave expansion (rapidly convergent)i,j(r,) = Plm () uij(r)Plm associated Legendre functions
Linear dependence always an issue
R-matrix wavefunctionkA i,jai,j,ki
Ni,jm bm,kmN+1
only represents the wavefunction within the R-matrix sphere
ai,j,kand bj,kvariationally determined coefficientsby diagonalising inner region secular matrix.Associated energy (“R-matrix pole”) is Ek.
kAkk
Full, energy-dependent scattering wavefunction given by
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-matricesAsymptotic boundary conditions:
Open channels
Closed channels
Defines the K (“reaction”)-matrix. K is real symmetric.Diagonalising K KD gives the eigenphase sum
Eigenphase sum
The K-matrix can be used to define the S (“scattering”)and T (“transition”) matrices. Both are complex.
, T = S 1
Propagate R-matrix(numerically v. stable)
Use eigenphase sumto fit resonances
S-matrices forTime-delays &MQDT analysis
Use T-matrices for total and differentialcross sections
UK R-matrix codes: www.tampa.phys.ucl.ac.uk/rmat
L.A. Morgan, J. Tennyson and C.J. Gillan, Computer Phys. Comms., 114, 120 (1999).
SCATCI:Special electronMolecule scatteringHamiltonian matrixconstruction
Processes: at low impact energies
Elastic scattering AB + e AB + e
Electronic excitation AB + e AB* + e
Dissociative attachment / Dissociative recombination AB + e A + B A + B
Vibrational excitation AB(v”=0) + e AB(v’) + e
Rotational excitation
AB(N”) + e AB(N’) + e
Impact dissociation
AB + e A + B + e
All go via (AB)** . Can also look for bound states
Adding photons
I. Weak fields
d = 4 2 a02 | <E
| m | 0>|2d
Atoms: Burke & Taylor, J Phys B, 29, 1033 (1975)Molecules: Tennyson, Noble & Burke, Int. J. Quantum Chem, 29, 1033 (1986)
II. Intermediate fieldsR-matrix – Floquet Method
Inner region Hamiltonian: linear molecule, parallel alignment
Colgan et al, J Phys B, 34, 2084 (2001)
III. Strong Fields
Time-dependent R-matrix method:
Theory: Burke & Burke, J Phys B, 30, L380 (1997)
Atomic implementation:
Van der Hart, Lysaght & Burke, Phys Rev A 76, 043405 (2007)
Atoms & molecules: numerical procedure
Lamprobolous, Parker & Taylor, Phys Rev A, 78, 063420 (2008)
Molecular implementation: awaited
Molecular alignment: Alex Harvey
• No Laser field.
• Re-scattering = scattering from molecular cation.
• Nuclei fixed at neutral molecules geometry
• Energy range up to ionisation potential of the cation (2nd
ionisation potential of the molecule)
• Relevant excited states included in calculations
• Initially looked at parallel alignment and elastic scattering.
• Total scattering symmetry 1Σg+ and 1Σu
+
Harvey & Tennyson J. Mod Opt. 54, 1099 (2007)Harvey & Tennyson J.Phys. B. 42, 095101 (2009)
Parallel Alignment
• Dipole selection rules and linearly polarised light. Molecule starts in ground state (aligned parallel to polarisation)
g1
u1 μ
g1
g1 μμ
• Odd number of photons 1Σg+ 1Σu
+ transition
• Even number of photons 1Σg+ 1Σg
+ transition
Results: H2 Total Cross Sections
• Close coupling expansion: 3 lowest ion states• Aligned xsec 3 to 4 times larger• Simplistic explanation: for 2Σg
+ ground state electron takes longer path through molecule – more chance to interact
H2 : Differential Cross Sections: low energies
Differential cross section against scattering angle (3-9eV)Top: Orientationally averaged, Bottom: parallel aligned, Left: 1Σg
+, Right: 1Σu+
•Energies below 1st resonance•1Σu
+: no shape change.•1Σg
+: Strong forward and backwards scattering
H2 : Differential Cross Sections: higher energies
•Energies above first resonance ~13eV and 1st threshold ~18eV.
•1Σg+ Inversion for DCS between first and second resonance.
Differential cross section against scattering angle (15.5-21eV)Left: 1Σg
+, Right: 1Σu+
H2 : Total Xsec as a function of alignment
Note: For non-parallel alignments need to consider other
symmetries but for 2Σg+ ground state we expect the contribution to
be minor.
Total cross section as a function of alignment angle β; Left: 1Σg
+, Right: 1Σu+
CO2 : Total xsec as a function of alignment• 3 state model: couples ion X, A and B states• Zero xsec for parallel alignment! Due to 2Πg ground state• Other symmetries important, e.g. expect Π total symmetry to be dominant at low energy as it couples to σ-continuum• 1Σg
+ peaks 50-55° deg and 125-130°•1Σu
+ peaks 90°, 35-40° and 140-145°
Total cross section as a function of alignment angle β
Experimental HHG intensity against alignment. Mairesse et al. J. Mod. Opt. 55 16 (2008)
1Σg+ 1Σu
+
Differential Cross Section: Polar Plots
Polar plots of the differential cross section as a function of scattering angle at 3eV, β = 30°; in the z-x plane; r = DCS, θ = scattering angle
Calculations neglect the Coulomb phase
1Σg+ 1Σu
+
Differential Cross Section: Polar Plots
Polar plots of the differential cross section as a function of scattering angle at 3 eV, β = 30°; in the z-x plane; r = DCS, θ = scattering angle
No Coulomb phase With Coulomb phase
1Σg+ 1Σu
+
Future Work
• Effect of adding more scattering symmetries.
• Inelastic cross sections.
• Extended energy range: RMPS method.
• Incorporate scattering amplitudes into a strong field model.
• Issues with treatment of Coulomb phases