The UK R-Matrix code

Department of Physics and AstronomyUniversity 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!But not for any molecule we want!And only in the gas phase!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

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

a = R-matrix radius normally set to 10 a0 (poly) and up to 20 a0 (diat)

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

R-matrix method

Two suites of codes, consisting of several modules (plenty of overlap) available:

diatomic: STOs and numerical integrationpolyatomic: GTOs and analytic integration

http://www.tampa.phys.ucl.ac.uk/rmat/

R-matrix suite

TARGET CALCULATION

INNER REGION CALCULATION

R-matrix suite

OUTER REGION CALCULATION

*

* INTERF in the diatomic case

Not very user Not very user friendly!friendly!

iNi,jci,jj= i,jci,j ║1 2 3… N ║

j N-electron configuration state function (CSF)

ci,jvariationally determined coefficients (standard diagonalisation techniques)

Target Wavefunctions

limit to number of configurations that can be includedlimit 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

ii,jai,j j=Molecular Orbitals

j: GTOs or STOs

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)

limit to number of basis functions that can be limit to number of basis functions that can be includedincluded

basis functions cannot be very diffusebasis 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 Problems with bigbig molecules due to computational limitations

ProblemsProblems with RydbergRydberg states (as they leak outside the box)

Inner region

kA i,j ai,j,ki

Ni,jjbj,kjN+1i

N= target states = CI target built in previous step

jN+1= L2 (integrable) functions

i,j = continuum orbitals = GTOs centred at CM or numerical

A Antisymmetrization operator

ai,j,kand bj,kvariationally determined coefficients

Full, energy-dependent scattering wavefunction given by:

kAkk

Inner region

kA i,j ai,j,ki

Ni,jjbj,kjN+1i

N= dictated by close-coupling

jN+1= dictated (not uniquely) by model used for target states

i,j = dictated by size of box and maximum Eke of scattering electron

ai,j,kand bj,kvariationally determined coefficients

limits size of box in polyatomic caselimits size of box in polyatomic case

limit to number of orbitals that can be includedlimit to number of orbitals that can be included

Choice of V0 does not have significant effect

Inner region

Inner region

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

Absolute energies do not matter;Everything depends on relative energies

E = 0

Outer region

i,j ai,j,kiNFj(rN+1) Ylm(N+1,N+1)r-1

N+1

Reduced radial functions Fj(rN+1) are single-centre.

Notice also there is no ANumber of angular behaviours to be include must be same as those included in inner region.

l l ≤ 6 (5 for polyatomic code)≤ 6 (5 for polyatomic code)

limit to number of channelslimit to number of channelsiNYlm(N+1,N+1)

Outer region

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 (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,....

• 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-)

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:

dRRRETRET viivviiv )();()()(

• 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,kk(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.

Electron impact dissociation(diatomics or pseudodiatomics)

ji

jillS

outinSjlil

in

ke

ke

EETSE

E

dE

d

2

3|),()12(|

4

Energy balance model within adiabatic nuclei approximation

Uses modified FN T-matrices

Neglected contributions of resonances

Cannot treat avoided crossings

<c (Eke , R)|Tvc (Ein , Eout , R)| v (R)>

Ein) d(Ein) d(Ein) d2(Ein)

dEout dddEout

R-matrix with pseudostates method (RMPS)

• inclusion ofiN 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!

kA i,j ai,j,ki

Ni,jjbj,kjN+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. C2

-) 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: