1. 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

2. 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

3. 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

4. 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 = a.u. (the average internuclear distance)
Largest calculations:
- 491-state, Nl =17-l, lmax = 3, projectile p.w. Lmax = 8
Convergence is established with:
state, Nl =15-l, lmax = 3, projectile p.w. Lmax = 8
state, , Nl =15-l, lmax = 2, projectile p.w. Lmax = 6
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)

5. 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

6. 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

7. e--H2 excitation ICS & DCS: B 1Sg+ - Lyman band
Oscillator Strength
Accurate theory Phys. Rev. A 60, 1226 (1999)
Fixed-nuclei CCC
Kato (measured) ± Phys. Rev. A 77, (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

8. 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)
Phys. Rev. A 95(2017)022708

9. 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, (2017)

10. 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

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

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

13. 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)

14. 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, ) 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

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 , unpublished

16. Electron-impact excitation from vibrational levels of H2

17. 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

18. 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

19. 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)
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)