Report

Uncertainties in calculations of low-energy resonant electron collisions with diatomic molecules Karel Houfek in collaboration with J. Horáček, M. Čížek and M. Formánek from Prague V. McKoy and C. Winstead from CalTech, USA J. Gorfinkiel and Z. Mašín from Open University, UK C.W. McCurdy and T. Rescigno, LBNL, USA Institute of Theoretical Physics Faculty of Mathematics and Physics Charles University in Prague Resonant electron-molecule collisions at low energies Vibrational (VE) and rotational excitation including elastic scattering e AB ( i , N i ) ( AB ) e AB ( f , N f ) Electronic excitation e AB ( AB ) e AB * Dissociative electron attachment (DA) e AB ( i ) ( AB ) A B three-body decays etc. We study also inverse process of DA called associative detachment (AD) A B ( AB ) e AB ( f ) 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 Motivation for this talk – results for e + CO SMC method for electron scattering + LCP and NRM for nuclear dynamics Fixed-nuclei eigenphase sums Potential energy curves Cross sections – vibrational excitation Motivation for this talk – results for e + CO SMC method for electron scattering + LCP and NRM for nuclear dynamics Cross sections – vibrational excitation 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 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 Another example of results – e + HCl Allan, Čížek, Horáček, Domcke, J. Phys. B 33 (2000) L209 HCl – structures in the VE and DA cross sections Nonlocal resonance model by Fedor et al – Phys. Rev. A 81 (2010) 042702 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 Uncertainties in calculations of e-M collisions 1) Potential energy curves (surfaces) 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 2) Fixed-nuclei electron scattering various models (SE, SEP, CAS) problem of consistency of scattering data with accurate potential energy curves 3) Model for nuclear dynamics different levels of approximation (effective range, LCP, NRM etc.) possibility of testing using two-dimensional model Potential energy curves – e + CO system Different methods with aug-cc-pVTZ basis Potential energy curves – e + CO system MRCI based on CASSCF(10,9) for larger basis sets LCP model with improved potentials – e + CO system Morse potential and adjusted width to get a good agreement with experiment 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 R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 9-12 virtual orbitals, 13 states at R = 2.1 R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals, 13 states R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals (1 different), 13 states R-matrix electron scattering calculations – CAS model CASSCF (10,8) model – 6-311G** basis, 11 virtual orbitals, 9 states Nuclear dynamics – local vs. nonlocal theory Vibrational excitation cross sections – e + CO Nuclear dynamics – local vs. nonlocal theory simple two-dimensional model as a testing tool Model Hamiltonian – one nuclear (R) and one electronic (r) degree of freedom H 1 d 2 2 dR 2 V0 ( R ) 1 d 2 dr 2 2 l ( l 1) 2r 2 V int ( R , r ) V0 (R) - potential energy of the neutral molecule - Morse potential l - angular momentum of the electron - p-wave (l = 1) or d-wave (l = 2) V int ( R , r ) - interaction potential - bound state of the electron for large R - resonance for small R Houfek, Rescigno, McCurdy, Phys. Rev. A 73 (2006) 032721 Houfek, Rescigno, McCurdy, Phys. Rev. A 77 (2008) 012710 Barrier for incoming electron → shape resonance for small R Fixed-nuclei calculations – N2-like model Electronic Hamiltonian used in fixed-nuclei calculations H el ( R ) 1 d 2 dr 2 r 2 (R) e 2 l ( l 1) 2r Cross sections (or phase shifts) 2 resonance position and width electron bounding energy Solution of the full 2D model Exact wave function at a given energy E E v E el i (R, r) (R, r) 0 1 E H i V int ( R , r ) ( R , r ) 0 with the initial state ( R , r ) vi ( R ) 0 2k rj l ( kr ) where v ( R ) is initial molecular vibrational state i N2-like model incoming electron Numerical solution using finite elements with DVR basis and exterior complex scaling NO-like model – scattered wave functions for vi = 0 Franck-Condon region Nuclear dynamics – LCP approximation Local complex potential approximation 2 1 d i (R) E E ( R ) ( R ) ( R ) res 2 E 2 2 2 dR 1/ 2 v (R) i Simple extensions of local complex potential approximation • barrier penetration factor, nonlocal imaginary part Details can be found in Trevisan et al, Phys. Rev. A 71 (2005) 052714 Vibrational excitation cross section Dissociative attachment cross section VE vi v f (E ) 4 3 2 ki 2 K DA v ( / 2 ) 1/ 2 f 2 NO-like model DA vi (E ) ki 2 lim d ( R ) R 2 E 2 Test of LCP approximation and its extensions NO-like model Nonlocal theory Direct derivation by choosing a proper diabatic basis for electronic part of the problem • discrete state d ( r ; R ) • othogonal “background” continuum states k ( r ; R ) satisfying conditions d (r , R ) R k (r , R ) 0, R lim d ( r ; R ) b ( r ) 0 R Into which we can expand the full wave function ( R , r ) d ( R ) d ( r , R ) kdk k ( R ) k ( r , R ) Using matrix elements of the electronic Hamiltonian in this basis V d ( R ) d H el ( R ) d , V dk ( R ) d H el ( R ) k V kk ' ( R ) k H el ( R ) k ' (V 0 ( R ) k / 2 ) ( k / 2 k ' / 2 ) 2 2 2 we finally get effective equations for nuclear motion E TR V d ( R ) d ( R ) F (E, R, R') dR ' F ( E , R , R ' ) d ( R ) V dk i ( R ) v i ( R ) dR ' kdkV dk ( R ) E T R V 0 ( R ) k / 2 i 2 1 V dk ( R ' ) Nonlocal theory – cross sections Vibrational excitation and dissociative attachment cross sections VE vi v f (E ) 4 2 ki 3 T VE vi v f 2 K DA 2 2 (E ) , DA vi (E ) ki 2 lim d ( R ) 2 R lim d ( r ; R ) b ( r ) It can be shown that for a properly chosen discrete state R there is no background contribution to the DA cross section. But the VE T-matrix consists of two terms T v i v f (E ) T v i v f (E ) T v i v f (E ) VE res bg The resonance term is calculated within nonlocal resonance theory Tv i v f (E ) v f V dk f d res R The background is non-zero even for inelastic vibrational excitation and for the 2D model can calculated exactly Tv i v f (E ) v f k f V int v i J k i v f V dk f J dk i v i bg J dk i ( R ) l l dr d ( r ; R ) J k i ( r ) * l l R Test of nonlocal theory – NO-like model smooth coupling (width) – works in all channels, reasonably small background Local vs. nonlocal theory – e + F2 model NO-like model – works in all channels, reasonably small background works in all channels, reasonably small background Minimizing background – e + F2 model the whole information about the dynamics is “hidden” in coupling (non-local potential) Conclusions Uncertainties in fixed-nuclei calculations 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 Uncertainties in nuclear dynamics 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 from model calculations