the r-matrix method for oriented molecules
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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
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