1. R-matrix calculations of electron molecule collisions
Jonathan Tennyson
University College London
Outer region
Inner region
ADAS Workshop
Armagh, Oct 2010

2. Processes: at low impact energies
Elastic scattering
AB + e AB + e
Electronic excitation
AB + e AB* + e
Vibrational excitation
AB(v”=0) + e AB(v’) + e
Rotational excitation
AB(N”) + e AB(N’) + e
Dissociative attachment / Dissociative recombination
AB + e A + B
A + B
Impact dissociation
AB + e A + B + e
All go via (AB-)** . Can also look for bound states

3. The R-matrix approach Inner region: Outer region: exchange
electron-electron correlation
multicentre expansion of 
Outer region:
exchange and correlation are negligible
long-range multipolar interactions are included
single centre expansion of 
Outer region e–
Inner region
C F C
R-matrix boundary r = a: target wavefunction = 0

4. R-matrix method for electrons: inner region wavefunction
(within the Fixed-Nuclei approximation)
Yk = A Si,j ai,j,k fiN hi,j + Si bj,k fjN+1
fiN= target states = CI target built from nuclear centred GTOs
fjN+1= L2 functions e-
inner region
hi,j = continuum orbitals =
GTOs centred on centre of mass
H H a
A = Anti-symmetriser
ai,j,k and bj,k variationally determined coefficients

5. Molecules of interest for fusion
Electron collisions with:
H2+
H2, D2, T2, HD, HT, DT C2
H2O
HCN/HNC
HCCH
In progress
CH4
CH+

6. Electron impact dissociation of H2
Important for fusion plasma and astrophysics
Low energy mechanism:
e- + H2(X 1Sg) e- + H2(b 3Su) e- + H + H
R-matrix calculations based on
adiabatic nuclei approximation
extended to dissociation

7. Determine choice of T-matrices to be averaged`
Including nuclear motion (within adiabatic nuclei approximation) in case of dissociation
Excess energy of incoming e- over dissociating energy can be split between nuclei and outgoing e- in any proportion.
Fixed nuclei excitation energy changes rapidly with bondlength
Tunnelling effects significant
ds(Ein)
dEout
Determine choice of T-matrices to be averaged

8. The energy balance method
D.T. Stibbe and J. Tennyson, New J. Phys., 1, 2 (1999).

9. Explicit adiabatic averaging over T-matrices using continuum functions

10. Need to Calculate: Total cross sections, s(Ein)
Energy differential cross sections, ds(Ein)
dEout
Angular differential cross sections, ds(Ein) dq
Double differential cross sections, d2s(Ein)
dqdEout
Required formulation of the problem
C.S. Trevisan and J. Tennyson, J. Phys. B: At. Mol. Opt. Phys., 34, 2935 (2001)

11. Integral cross sections
e- + H e- + H + H
Integral cross sections
Cross section (a02)
Incoming electron energy (eV)

12. e- + H2(v=0) e- + H + H Energy differential cross sections in a.u.
Atom kinetic energy (eV)
Incoming electron energy (eV)

13. e- + H2(v>0) e- + H + H Energy differential cross sections in a.u.
Atom kinetic energy (eV)
Incoming electron energy (eV)

14. e- + H2 e- + H + H Energy differential cross sections
as a function of incoming electron energy
Atom kinetic energy (eV)
Extended to D2, T2, mixed isotopologues,
C.S. Trevisan & J. Tennyson, Plasma Phys. Controlled Fusion, 44, 1263 (2002)

15. Electron – water rotationally resolved cross sections:
Differential cross sections (DCS) at 6 eV
Cho et al (2004) *
Jung et al (1982)
DJ=1
DJ=all
DJ=0

16. Electron – water (rotationally averaged) elastic cross sections
Integral cross section
A Faure, JD Gorfinkel & J Tennyson J Phys B, 37, 801 (2004)

17. G. Halmova, JD Gorfinkel & J Tennyson J Phys B, 39, 2849 (2006).
Electron – C2:
C2 states
G. Halmova, JD Gorfinkel & J Tennyson J Phys B, 39, 2849 (2006).

18. Uracil
Electron capture mainly due to DNA and RNA bases, because these have extended aromatic system
Scattering electron can temporary be attached to an unocuppied π* orbital, giving rise to a shape resonance

19. Shape resonances in e- - uracil
Positions (and widths) in eV of 2A” resonances
cc-pVDZ, a=13 a0, 15 virtuals, CAS(14,10)
Method
Res 1
Res 2
Res 3
SEP SE
2.25 (0.26)
4.43 (0.41)
8.62 (2.69)
0.31 (0.015)
2.21 (0.16)
5.21 (0.72) CC
0.134 (0.0034)
1.94 (0.168)
4.94 (0.38)
Obs
0.22
1.58
3.83
A Dora, J Tennyson, L Bryjko and T van Mourik
J Chem Phys, 130, (2009)

20. Resonances in electron - uracil: Shape and Feshbach
Positions (and widths) in eV of 2A’ and 2A” resonances
CC, cc-pVDZ, a=13 a0, 15 virtuals, CAS(14,10)
Symmetry
Res 1
Res 2
Res 3
2A’
6.17 (0.15)
7.62 (0.11)
8.12 (0.14)
2A”
0.134 (0.0034)
1.94 (0.168)
4.94 (0.38)

21. Total elastic cross section

22. Elastic DCS (differential cross section) at 90º

23. Electron impact electronic excitation cross section
X 1A’
3 1A’
with Born correction
without Born correction
2 3A’

24. UK molecular R-matrix codes:
General and flexible for electron-molecule collisions
Treat a variety (all) low-energy processes
Particularly good for ions
New R-matrix with pseudo-state (RMPS) leads to greater accuracy and extended energy range
J. Tennyson,
Electron - molecule collision calculations using the R-matrix method,
Phys. Rep., 491, 26 (2010).

25. “The best book for anyone who is embarking on research in
“The best book for anyone who is
embarking on research in
astronomical spectroscopy”
Contemporary Physics (2006)