Karel Houfek

1. Uncertainties in calculations of low-energy resonant electron collisions with diatomic molecules KarelHoufek in collaborationwithJ. Horáček, M. Čížek and M. Formánek fromPragueV. McKoy and C. Winstead from CalTech, USAJ. Gorfinkiel and Z. Mašínfrom Open University, UKC.W. McCurdyand T. Rescigno, LBNL, USA Institute of Theoretical Physics Faculty of Mathematics and Physics Charles University in Prague

2. Resonant electron-molecule collisions at low energies Vibrational (VE) and rotationalexcitationincluding elastic scattering Electronic excitation Dissociative electron attachment (DA) three-body decays etc. We study also inverse process of DA called associative detachment (AD)

3. Theoretical description of electron-molecule collisions • First fixed-nuclei calculations provide • potential energy curves (surfaces) of neutral molecule (standard)and molecular ion where electron is bound (more difficult) • fixed-nuclei scattering data (eigenphase sums, cross sections) –several methods available • Complex Kohn variational principle (Berkeley, USA) • R-matrix (UCL and OU, UK) • Schwinger multichannel variational method (CalTech, USA) • from which a model for nuclear dynamics is constructed within some approximation • local complex potential approximation – simplest to use, first choice • nonlocal, complex, and energy-dependent potential – universal, but difficult • R-matrix approach of Schneider et al – applied only to N2 and CO

4. Motivationforthistalk – results for e + COSMC methodfor electron scattering + LCP and NRM for nuclear dynamics • Fixed-nuclei eigenphase sums • Cross sections – vibrational excitation • Potential energy curves

5. Motivationforthistalk – results for e + COSMC methodfor electron scattering + LCP and NRM for nuclear dynamics • Cross sections – vibrational excitation

6. e + CO – comparison with previous calculations • What is the best theoretical result and what uncertainties are there? • Morgan, J. Phys. B 24 (1991) 4649 • Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005

7. e + CO – comparison of potential energy curves • This gives us a hint of the origin of discrepancies and uncertainties • Morgan, J. Phys. B 24 (1991) 4649 • Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005

8. Another example of results – e + HCl • Allan, Čížek, Horáček, Domcke, J. Phys. B 33 (2000) L209

9. HCl – structures in the VE and DA cross sections Nonlocal resonance model by Fedoret al – Phys. Rev. A81 (2010) 042702

10. HCl – origin of structures in the cross sections • Uncertainties in position of structures – shape and relative position of PEC • Uncertainties in absolute values of the cross sections – details of the model used

11. Uncertainties in calculations of e-M collisions • 1) Potential energy curves (surfaces) • 2) Fixed-nuclei electron scattering • 3) Model for nuclear dynamics • various methods (HF, CASSCF, CI, CC) • absolute energies are not important • relative shape and position of curves (surfaces) is crucial, problem of size consistency (N and N+1 electrons) • known difficulties to obtain correct electron affinities • various models (SE, SEP, CAS) • problem of consistency of scattering data with accurate potential energy curves • different levels of approximation (effective range, LCP, NRM etc.) • possibility of testing using two-dimensional model

12. Potential energy curves – e + CO system • Different methods with aug-cc-pVTZ basis

13. Potential energy curves – e + CO system • MRCI based on CASSCF(10,9) for larger basis sets

14. LCP model with improved potentials – e + CO system • Morse potential and adjusted width to get a good agreement with experiment

15. R-matrix electron scattering calculations • Schwinger multichannel variational method cannot describe the target beyond the Hartree-Fock approximation –> impossible to have a better description of the target consistent with electron scattering calculations • Possible with UK R-matrix polyatomic codes – several different scattering models available • SE – static exchange – target at HF level • SEP – static exchange plus polarization – target at HF level • CAS – close-coupling model (only a few excited target states included) based on complete active space (CASSCF) calculations of the target

16. R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 9-12 virtual orbitals, 13 states at R = 2.1

17. R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals, 13 states

18. R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals (1 different), 13 states

19. R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 11 virtual orbitals, 9 states

20. Nuclear dynamics – local vs. nonlocal theory • Vibrational excitation cross sections – e + CO

21. Nuclear dynamics – local vs. nonlocal theorysimple two-dimensional model as a testing tool Model Hamiltonian – one nuclear (R) and one electronic (r) degree of freedom incoming electron vibrational motion - potential energy of the neutral molecule- Morse potential - angular momentum of the electron- p-wave (l = 1) or d-wave (l = 2) - interaction potential- bound state of the electron for large R- resonance for small R Barrier for incoming electron → shape resonance for small R Houfek, Rescigno, McCurdy, Phys. Rev. A 73 (2006) 032721Houfek, Rescigno, McCurdy, Phys. Rev. A 77 (2008) 012710

22. Fixed-nuclei calculations – N2-like model Electronic Hamiltonian used in fixed-nuclei calculations Cross sections (or phase shifts) resonance position and width electron bounding energy

23. Solution of the full 2D model Exact wave function at a given energy with the initial state where is initial molecular vibrational state Numerical solution usingfinite elements with DVR basisand exterior complex scaling N2-like model incoming electron vibrational motion

24. NO-like model – scattered wave functions for vi = 0 Franck-Condon region

25. Nuclear dynamics – LCP approximation Local complex potential approximation • Simple extensions of local complex potential approximation • barrier penetration factor, nonlocal imaginary part • Vibrationalexcitationcrosssection • Dissociative attachmentcrosssection Details can be found in Trevisanet al,Phys. Rev. A 71 (2005) 052714 NO-like model

26. Test of LCP approximation and its extensions NO-like model

27. Nonlocal theory • Direct derivation by choosing a proper diabatic basis for electronic part of the problem • discrete state • othogonal “background” continuum states • satisfying conditions • Into which we can expand the full wave function • Using matrix elements of the electronic Hamiltonian in this basis • we finally get effective equations for nuclear motion

28. Nonlocal theory – cross sections Vibrational excitation and dissociative attachment cross sections It can be shown that for a properly chosen discrete statethere is no background contribution to the DA cross section. But the VE T-matrix consists of two terms The resonance term is calculated within nonlocal resonance theory The background is non-zero even for inelastic vibrational excitationand for the 2D model can calculated exactly

29. Test of nonlocal theory – NO-like model smooth coupling (width) – works in all channels, reasonably small background

30. Local vs. nonlocal theory – e + F2 model works in all channels, reasonably small background NO-like model – works in all channels, reasonably small background

31. Minimizing background – e + F2 model the whole information about the dynamics is “hidden”in coupling (non-local potential)

32. Conclusions • Uncertainties in fixed-nuclei calculations • Uncertainties in nuclear dynamics • shape of potential energy curves (surfaces), relative positions of potentials for neutral molecule and molecular negative ion– advanced quantum chemistry methods (MRCI, CCSD(T) etc) are necessary, basis sets limit, no fitting to Morse or similar analytical potential –> comparison with available experimental data (electron affinities, spectroscopic constants etc.) • problem of consistency of scattering data with accurate potential energy curves– it is necessary to go beyond HF description of the target, adjusting parameters of electron scattering calculations to get correct electronic energies where electron is bound –> estimating errors by comparison of several scattering models • nonlocal theory necessary in many cases, simple extensions of local complex potential approximation can sometimes improved the results, but it strongly depends on the system • unknown background contribution –> uncertainties estimates frommodel calculations