Curtin University, Perth, Australia Adiabatic nuclei calculations of electron and positron scattering from molecular hydrogen and its ion D.V. Fursa Curtin University, Perth, Australia in collaboration with: M.C. Zammit, J.S. Savage, L.H. Scarlett, J. Tapley, I. Bray Australian Research Council Curtin University Los Alamos National Laboratory US Air Force Office of Scientific Research Pawsey Supercomputing Centre Acknowledgements

Overview Fixed-nuclei (FN) CCC method Application to e-H2 scattering convergence studies cross section results stopping power Adiabatic-nuclei (AN) CCC method low energy b 3Su+ state excitations of H2 Spheroidal coordinate formulation of the CCC dissociation cross sections electronic excitation radiative decay (EV) processes Conclusions

Fixed-nuclei (FN) CCC method Born-Oppenheimer approximation Fixed-nuclei approximation, R = fixed Solve for the electronic wave function Target Hamiltonian HT is diagonalized in a Sturmian (Laguerre) basis modeling of infinite number of bound and continuum states with a finite number of pseudostates Single-center spherical coordinate formulation Spheroidal coordinate formulation N-state multi-channel expansion The Schrödinger equation is converted to an integral LS equation for the T-matrix Solved by projectile partial-wave expansion Cross sections

Electron scattering from H2 Is a benchmark system with a range of applications Accurate measurements exist for some major processes Theory had not done sufficiently well: Previous largest CC calculations use 9-states RM and SMC or 41-states RMPS There is room for improvement… e--H2 recommended cross sections of Yoon et al. J. Phys. Chem. Ref. Data 37(2008)913 are all derived from experimental data Five models : single-center spherical coordinate formulation Fixed-nuclei calculations at R = 1.448 a.u. (the average internuclear distance) Largest calculations: - 491-state, Nl =17-l, lmax = 3, projectile p.w. Lmax = 8 Convergence is established with: - 427-state, Nl =15-l, lmax = 3, projectile p.w. Lmax = 8 - 259-state, , Nl =15-l, lmax = 2, projectile p.w. Lmax = 6 - 92-state: only negative energy (bound) states from 491 state model - 9-state: first 9 states of H2 Good convergence when CCC(491) = CCC(427) Spheroidal coordinate formulation: same results as CCC(491)

Cross sections: for electron scattering from H2 The best estimate of CCC cross sections is produced for energies from 0.1 to 300 eV for total cross section total ionization cross section elastic scattering (ICS, DCS) excitation cross sections (ICS, DCS) for b 3Su+ , a 3Sg+, c 3Pu , e 3Su+, h 3Sg+ B 1Su+, C 1Pu, EF 1Sg+, B’ 1Su+, D 1Pu, B” 1Su+, D’ 1Pu, H 1Sg+, GK 1Sg+, I 1Pu, , J 1Dg more transitions are available on request Results available from LXcat and ALADDIN databases B 1Su+ : Lyman band C 1Pu : Werner band

e--H2 excitation ICS & DCS: b 3Su+ SMC, RM ~ 9-state recommended cross sections are due to Yoon et al., J. Phys. Chem. Ref. Data 37(2008)913 Zammit, Savage, Fursa & Bray, Phys. Rev. A 95(2017)022708

e--H2 excitation ICS & DCS: B 1Sg+ - Lyman band Oscillator Strength Accurate theory 0.274 Phys. Rev. A 60, 1226 (1999) Fixed-nuclei CCC 0.288 Kato (measured) 0.241 ± 0.048 Phys. Rev. A 77, 062708 (2008) SMC, RM ~ 9-state recommended cross sections are due to Yoon et al., J. Phys. Chem. Ref. Data 37(2008)913 Zammit, Savage, Fursa & Bray, Phys. Rev. A 95(2017)022708

e--H2 total ionization cross section recommended cross sections are due to Yoon et al., J. Phys. Chem. Ref. Data 37(2008)913 Zammit, Savage, Fursa & Bray, Phys. Rev. Lett. 116 (2016) 233201 Phys. Rev. A 95(2017)022708

Electron Mass Stopping Powers in H2 Mean excitation energy Previous estimates of the stopping power: phenomenological extension of the Bethe formula to low and intermediate energies evaluation of the available experimental and theoretical data for excitation and ionization cross sections. Fursa et al., Phys. Rev. A 96, 022709 (2017)

Adiabatic-nuclei (AN) CCC method AN T-matrix: transition to the vibrational level m of the electronic state n from initial electronic state i and vibrational state n Vibrational wave functions fnm(R) are obtained by diagonalizing the Born-Oppenheimer Hamiltonian Vibrationally resolved cross sections The closure property Cross sections summed over the final electronic level vibrational states FN approx.  The AN CCC method was applied to: Electron collisions with hot (vibrationally excited) H2+ and isotopologues, and Kinetic Energy Release of Fragments, PRA 96(2017)022706, PRA 90(2014)022711 Positron-H2 low-energy collisions, vibrational v=0  v=1 excitation of H2, PRA 95(2017)022707

e--H2 excitation Low energy b 3Su+ AN DCS & ICS FN excitation threshold: 10.31 eV Fursa et al., submitted to PRA

AN DCS & ICS: isotope effect e--H2 excitation Low energy b 3Su+ AN DCS & ICS: isotope effect Fursa et al., submitted to PRA

CCC: spheroidal coordinate formulation Natural choice for diatomic molecules For H2+: separation of variables Spheroidal coordinates: Target Hamiltonian HT is diagonalized in a Sturmian (Laguerre) basis Models & convergence 251 states 27 states Spherical CCC(491) + dissociation fractions CC(27)

Electron-impact dissociation of H2 to neutral fragments Spherical CCC(491) + dissociation fractions CC(27) Recommended cross sections are due to Yoon et al. J.Phys.Chem.Ref.Data 37(2008)913 based on Corrigan (J. Chem. Phys. 43, 4381 1965) data Main features low energy dominated by b 3Su+ state other triplet states make equally large contribution at the peak and above singlet states dominate above 50 eV : For singlet states it is hard to make an accurate estimate: error 6 – 14 % Possible dissociation channels: dissociative excitation predissociation excitation radiative decay dissociation At high energies (> 50 eV) the main dissociation pathway for H2 is radiative decay to the ground state vibrational continuum. Fursa et al., submitted to EPJD

Electron-impact dissociative excitation of H2 Fursa et al. , unpublished

Electron-impact excitation from vibrational levels of H2

Electronic excitation radiative decay (EV) cross sections X 1Sg+(v=0)  B 1Su+, C 1Pu, E,F 1Sg+, D 1Pu, B’ 1Su+ X 1Sg+(v”=0) X 1Sg+(v=0)  C 1Pu, B 1Su+ X 1Sg+(v') Fursa et al., unpublished

Conclusions Thank you First large-scale close-coupling calculations for H2 First comprehensive theoretical dataset of e-H2 excitation cross sections Uncertainties: < 11%, better than 5% for most transitions First ab initio estimates of stopping power in H2 Adiabatic nuclei approach & spheroidal coordinate formulation of the CCC - electron collisions with hot (vibrationally excited) H2+ and isotopologues, and Kinetic Energy Release of Fragments - positron-H2 low-energy collisions, vibrational v=0  v=1 excitation of H2 - low energy b 3Su+ state excitation of H2 - dissociation of H2 - electronic excitation radiative decay (EV) processes Results available from LXcat and ALADDIN databases

Adiabatic-nuclei (AN) CCC method: e--H2+ Electron scattering from hot (vibrationally excited) H2+ with results for dissociative excitation ionization and proton production cross sections obtained for H2+ and its istotopologues: H2+, D2+, T2+, HD+, HT+, DT+. Zammit et al., Phys. Rev. A 90 (1014) 022711 Results available from LXcat and ALADDIN databases Kinetic Energy Release of Fragments from Electron-Impact Dissociation of the Molecular Hydrogen Ion and its Isotopologues Scarlett et al., Phys. Rev. A 96(2017) 022706