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The UK R-Matrix code
Jimena D. Gorfinkiel
Department of Physics and Astronomy
University College London
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
Fixed-nuclei approximation: nuclei are held fixed during the
collision, i.e., nuclear motion is neglected
R-matrix method for electron-molecule collisions
Inner region:
einner region
C
a
outer region
a = R-matrix radius normally set to
10 a0 (poly) and up to 20 a0 (diat)
• 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
R-matrix method
1. calculation of target properties: electronic energies and
transition moments
2. inner region:
calculation of k from 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
R-matrix suite
INNER REGION CALCULATION
TARGET CALCULATION
http://www.tampa.phys.ucl.ac.uk/rmat/
R-matrix suite
*
* INTERF in
the diatomic
case
Not very user
friendly!
OUTER REGION CALCULATION
Target Wavefunctions
Configuration interaction calculations
fiN = Si,j ci,jzNj= Si,j ci,j ║1 2 3… N ║
zNj N-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,j variationally determined coefficients (standard
diagonalisation techniques)
Target Wavefunctions
i = Si,j ai,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)
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)
Inner region
k = A Si,j ai,j,k fiN  i,j + Sj bj,k fjN+1
fiN= target states = CI target built in previous step
fjN+1= L2 (integrable) functions
 i,j = continuum orbitals = GTOs centred at CM or numerical
A
Antisymmetrization operator
ai,j,k and bj,k variationally determined coefficients
Full, energy-dependent scattering wavefunction given by:
Y(E) = Sk Ak(E) k
Inner region
k = A Si,j ai,j,k fiN i,j + Sj bj,k fjN+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,k and bj,k variationally determined coefficients
Inner region
Choice of V0 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
Excited states
Ground state
Target state energies
N+1 electron states
‘Continuum states’
(only discretised in
the R-matrix method)
E=0
Bound states of the
compound system
Absolute energies do not matter;
Everything depends on relative energies
Outer region
Y= Si,j ai,j,k fiN Fj(rN+1) Ylm(N+1,fN+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 channels fiN Ylm(N+1,fN+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,....
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?
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
Fails at very low energy
Fails in the presence of resonances
* Gianturco and Jain, Phys. Rep. 143 (1986) 347
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:
Tiviv ( E ) =   v ( R)Tii ( E; R)  v ( R)dR
• 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 nonadiabatic couplings is needed)
• Incorporates effect of resonances
• Narrow avoided crossing must be diabatized
k= Si,j i,j,k k(R0) z(R)
z 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
2
 (Ein) d(Ein) d(Ein) d (Ein)
dEout
dQ
dQdEout
 Neglected contributions of resonances
 Cannot treat avoided crossings
d
 Eke
=
3
dEke 4 Ein
| (2S + 1)T


S li l j
S
ili jl j
( Ein , Eout ) |
<Xc (Eke , R)|Tvc (Ein , Eout , R)| Xv (R)>
2
R-matrix with pseudostates method (RMPS)
Yk = A Si,j ai,j,k fiN hi,j + Sj bj,k fjN+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)
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
Some bibliography: