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