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

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!

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 nuclear motion is neglected Fixed-nuclei approximation: nuclei are held fixed during the collision, i.e., nuclear motion is neglected

R-matrix method for electron-molecule collisions C 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 a inner region outer region e-e- a = R-matrix radius normally set to 10 a 0 (poly) and up to 20 a 0 (diat)

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

R-matrix suite TARGET CALCULATION INNER REGION CALCULATION

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

 i N  i,j  c i,j   j =  i,j  c i,j ║  1  2  3…  N ║   j N-electron configuration state function (CSF) c i,j  variationally determined coefficients (standard diagonalisation techniques) Target Wavefunctions limit to number of configurations that can be included Configuration interaction calculations Models used: CAS (most frequent), CASSD,single configuration, etc… Inner shells normaly frozen

Target Wavefunctions  i  i,j  a i,j  j =  Molecular Orbitals  j : GTOs or STOs a i,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) limit to number of basis functions that can be included basis functions cannot be very diffuse

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 big Problems with big molecules due to computational limitations ProblemsRydberg Problems with Rydberg states (as they leak outside the box)

Inner region  k   A  i,j a i,j,k  i N  i,j  j  b j,k  j N+1  i N = target states = CI target built in previous step  j N+1 = L 2 (integrable) functions  i,j = continuum orbitals = GTOs centred at CM or numerical A Antisymmetrization operator a i,j,k  and b j,k  variationally determined coefficients Full, energy-dependent scattering wavefunction given by:  k  A k  k

Inner region  k   A  i,j a i,j,k  i N  i,j  j  b j,k  j N+1  i N = dictated by close-coupling  j N+1 = dictated (not uniquely) by model used for target states  i,j = dictated by size of box and maximum E ke of scattering electron a i,j,k  and b j,k  variationally determined coefficients limits size of box in polyatomic case limit to number of orbitals that can be included

Choice of V 0 does not have significant effect Inner region

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

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)

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

Outer region    i,j a i,j,k  i N  F j (r N+1 ) Y lm (  N+1,  N+1 )r -1 N+1 Reduced radial functions F j (r N+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 channels  limit to number of channels  i N  Y lm (  N+1,  N+1 )

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

Processes we can study Rotational excitation for diatomics and triatomics (H 2, H 3 +, H 2 O, etc.) Vibrational excitation for diatomics (e.g. HeH + ) Electron impact dissociation for H 2 (and 1-D for H 2 O) Provide resonance information for dissociative recombination studies (CO 2+, HeH +, NO + ) Elastic collisions* Electronic excitation* * for ‘reasonable-size’ molecules: H 2 O, NO, N 2 O, H 3 +, CF, CF 2, CF 3, OClO, Cl 2 O, SF 2,....

Collisions with bigger molecules (C 4 H 8 O) Intemediate energies and in particular ionisation (low for certain systems) Full dimension DEA study of H 2 O Collisions with negative ions (C 2 - ) Processes we have recently started studying Need to re-think some of the strategies? Program upgrade?

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 l contribution :  calculated using Coulomb-Born approximation * Gianturco and Jain, Phys. Rep. 143 (1986) 347 Fails at very low energy Fails in the presence of resonances

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

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     i,j  i,j,k  k (R 0 )  (R)  are Legendre polinolmials and  i,j,k are obtained  diagonalising the total H Lots of hard work, particularly to untangle curves. Rather crude approximation as lots of R dependences are neglected.

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  E in ) d  (E in ) d  (E in ) d 2  (E in ) dE out d  d  dE out

R-matrix with pseudostates method (RMPS) inclusion of  i N that are not true eigenstates of the system to represent discretized continuum: “pseudostates” transitions to pseudostates are taken as ionization (projection may be needed) 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!  k   A  i,j a i,j,k  i N  i,j  j  b j,k  j N+1

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. C 2 - ) that cannot presently be addressed Molecular RMPS method useful for: * J. D. Gorfinkiel and J. Tennyson, J. Phys. B 38 (2004) L 321

Some bibliography: