The UK R-Matrix code
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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:
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