The UK R-Matrix code

1. TheUK R-Matrix code Jimena D. Gorfinkiel Department of Physics and Astronomy University College London

2. What processes can we treat? • LOW ENERGY: rotational, vibrational and electronic excitation • INTERMEDIATE ENERGY: electronic excitation and ionisation But not for any molecule we want! And only in the gas phase!

3. R-matrix method • Used (these days) mostly to treat electronic excitation • Nuclear motion can be treated within adiabatic approximations (for rotational OR vibrational motion) • Non-adiabatic effects have been included in calcualtions for diatomics • All (so far) imply running FIXED-NUCLEI calculations Fixed-nuclei approximation: nuclei are held fixed during the collision, i.e., nuclear motion is neglected

4. R-matrix method for electron-molecule collisions Inner region: • exchange and correlation important • multicentre expansion • adapt quantum chemistry techniques Outer region: • exchange and correlation are negligible • long-range multipolar interactions sufficient • single centre expansion • adapt atomic R-matrix codes e- C inner region a outer region a = R-matrix radius normally set to 10 a0 (poly) and up to 20 a0 (diat)

5. R-matrix method 1. calculation of target properties: electronic energies and transition moments 2. inner region: calculation of kfrom diagonalization of HN+1 3. outer region: match channels at the boundary and propagate the R-matrix to the asymptotic limit Two suites of codes, consisting of several modules (plenty of overlap) available: diatomic: STOs and numerical integration polyatomic:GTOs and analytic integration

6. INNER REGION CALCULATION TARGET CALCULATION R-matrix suite http://www.tampa.phys.ucl.ac.uk/rmat/

7. R-matrix suite * * INTERF in the diatomic case Not very user friendly! OUTER REGION CALCULATION

8. Target Wavefunctions Configuration interaction calculations fiN= Si,jci,jzNj=Si,jci,j║1 2 3… N ║ zNjN-electron configuration state function (CSF) limit to number of configurations that can be included Models used: CAS (most frequent), CASSD,single configuration, etc… Inner shells normaly frozen ci,jvariationally determined coefficients (standard diagonalisation techniques)

9. Target Wavefunctions i= Si,jai,jj=Molecular Orbitals j: GTOs or STOs limit to number of basis functions that can be included basis functions cannot be very diffuse ai,j can be obtained in a variety of ways: • SCF Hartree-Fock • Diagonalisation of the density matrices  Pseudo-natural orbitals • Other programs (CASSF in MOLPRO)

10. Target Wavefunctions Eigenvectors and eigenvalues are determined and the transition moments are obtained from the density matrices Quality of representation is very good for 2/3 atom molecules Problems with big molecules due to computational limitations Problems with Rydberg states (as they leak outside the box)

11. Inner region k=ASi,j ai,j,kfiNi,j+ Sjbj,kfjN+1 fiN= target states = CItarget built in previous step fjN+1= L2(integrable) functions i,j = continuum orbitals= GTOscentred at CM or numerical A Antisymmetrization operator ai,j,kand bj,kvariationally determined coefficients Full, energy-dependent scattering wavefunction given by: Y(E) = SkAk(E)k

12. Inner region k=ASi,j ai,j,kfiNi,j+ Sjbj,kfjN+1 fiN= dictated by close-coupling fjN+1= dictated (not uniquely) by model used for target states i,j = dictated by size of box and maximum Eke of scattering electron limits size of box in polyatomic case limit to number of orbitals that can be included ai,j,kand bj,kvariationally determined coefficients

13. Inner region Choice of V0 does not have significant effect

14. Inner region In spite of orthogonalisation, linear dependence can be serious problem  limit to quality of continuum representation

15. Inner region Two diagonalisation alternatives: Givens-Housholder method or recently implemented Partitioned R-matrix (a few of the poles are calculated using Arnoldi method and the contribution of the rest is added as a correction)

16. Scattering wavefunction: the need for balance N-electron statesN+1 electron states ‘Continuum states’ (only discretised in the R-matrix method) Excited states Ground state E = 0 Bound states of the compound system Target state energies Absolute energies do not matter; Everything depends on relative energies

17. Outer region Y=Si,j ai,j,kfiNFj(rN+1) Ylm(N+1,N+1)r-1N+1 Reduced radial functions Fj(rN+1)are single-centre. Notice also there is no A Number of angular behaviours to be include must be same as those included in inner region. l ≤ 6 (5 for polyatomic code) limit to number of channelsfiNYlm(N+1,N+1)

18. Outer region

19. Outer region • Using information form the inner region and the target calculation (to define the channels) the R-matrix at the boundary is determined. • The R-matrix is propagated and matched to analytic asymptotic functions. • At sufficiently large distances K-matrices are determined using asymptotic expressions • Diagonalizing K-matrices we can find resonance positions and widths • From K-matrices we can obtain T-matrices and cross sections

20. Processes we can study • Rotational excitation for diatomics and triatomics (H2, H3+, H2O, etc.) • Vibrational excitation for diatomics (e.g. HeH+) • Electron impact dissociation for H2 (and 1-D for H2O) • Provide resonance information for dissociative recombination studies (CO2+, HeH+, NO+) • Elastic collisions* • Electronic excitation* * for ‘reasonable-size’ molecules: H2O, NO, N2O, H3+, CF, CF2, CF3 , OClO, Cl2O, SF2,....

21. Processes we have recently started studying • Collisions with bigger molecules (C4H8O) • Intemediate energies and in particular ionisation (low for certain systems) • Full dimension DEA study of H2O • Collisions with negative ions (C2-) Need to re-think some of the strategies? Program upgrade?

22. Rotational excitation (Alexandre Faure, Observatoire de Grenoble) • Adiabatic-nuclei-rotation (ANR) method (Lane, 1980) • Applied to linear and symmetric top molecules • Low l contribution:  calculated from BF FN T-matrices obtained from R-matrix calculations • High lcontribution:  calculated using Coulomb-Born approximation Fails at very low energy Fails in the presence of resonances * Gianturco and Jain, Phys. Rep. 143 (1986) 347

23. Vibrational excitation (not used for 5 years, Ismanuel Rabadan) • Adiabatic model (Chase, 1956) • Using fixed-nuclei T-matrices and vibrational wavefunctions obtained by solving the Schrodinger equation numerically: • used for low v • limitations same as before

24. Non-adiabatic effects (not used for 5 years, Lesley Morgan) • Provides vibrationally resolved cross sections • Couples nuclear and electronic motion (no calculation of non-adiabatic couplings is needed) • Incorporates effect of resonances • Narrow avoided crossing must be diabatized k=Si,ji,j,kk(R0)(R)  are Legendre polinolmials and i,j,k are obtaineddiagonalising the total H Lots of hard work, particularly to untangle curves. Rather crude approximation as lots of R dependences are neglected.

25. s(Ein) ds(Ein) ds(Ein) d2s(Ein) dEout dQ dQdEout Electron impact dissociation (diatomics or pseudodiatomics) • Energy balance model within adiabatic nuclei approximation • Uses modified FN T-matrices • Neglected contributions of resonances • Cannot treat avoided crossings <Xc (Eke , R)|Tvc (Ein , Eout , R)| Xv (R)>

26. R-matrix with pseudostates method (RMPS) Yk=ASi,j ai,j,kfiNhi,j+ Sjbj,kfjN+1 • inclusion of fiN that are not true eigenstates of the system to represent discretized continuum: “pseudostates” • obtained by diagonalizing target H • must not (at least most of them) represent bound states • In practice: inclusion of a different set of configurations and another basis set (on the CM); problems with linear dependence! • transitions to pseudostates are taken as ionization (projection may be needed)

27. Molecular RMPS method useful for: • Extending energy range of calculations • Treating near threshold ionization • Improving representation of polarization (very important at low energies but difficult to achieve without pseudostates) • Will also allow us to treat excitation to high-lying electronic states and collisions with anions (e.g. C2-) that cannot presently be addressed * J. D. Gorfinkiel and J. Tennyson, J. Phys. B 38 (2004) L 321

28. Some bibliography: