R-matrix theory and Electron-molecule scattering

1. 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

2. 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 b=0.

3. 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).

4. Wigner-Eisenbud R-matrix theory Outer region e- H H Inner region R-matrix boundary

5. 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

6. Eq. 1 Expand u in terms of basis functions v Coefficients aijkdetermined by solving Inserting this into eq. 1 Eq. 2

7. 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.

8. 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.

9. 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.

10. Electron – molecule collisions Outer region e- H H Inner region R-matrix boundary

11. 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

12. Inner region: Scattering wavefunctions Yk= A Si,jai,j,k fiN hi,j + Sm bm,k fmN+1 where fiN N-electron wavefunction of ith target state hi,j1-electron continuum wavefunction fmN+1 (N+1)-electron short-range functions ‘L2’ ai,j,kand bj,kvariationally determined coefficients A Antisymmetrizes the wavefunction

13. Target Wavefunctions fiN = Si,jci,jzj where fiN N-electron wavefunction of ith target state zjN-electron configuration state function (CSF) Usually defined using as CAS-CI model. Orbitals either generated internally or from other codes ci,jvariationally determined coefficients

14. 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 • Polyatomic code: l < 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

15. R-matrix wavefunction Yk= A Si,jai,j,k fiN hi,j + Smbm,k fmN+1 only represents the wavefunction within the R-matrix sphere ai,j,kand bj,kvariationally determined coefficients by diagonalising inner region secular matrix. Associated energy (“R-matrix pole”) is Ek. Full, energy-dependent scattering wavefunction given by Y(E) = SkAk(E) Yk Coefficients Ak determined in outer region (or not) Needed for photoionisation, bound states, etc. Numerical stability an issue.

16. 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

17. 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).

18. 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

19. 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 + B Impact dissociation AB + e A + B + e All go via (AB-)** . Can also look for bound states

20. Electron - LiH scattering:2S eigenphase sums

21. 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