B-spline basis sets in exterior complex scaling for single and double photoionization studies

1. B-spline basis sets in exterior complex scaling for single and double photoionization studies Fernando Martín Departamento de Química, Universidad Autónoma de Madrid Spain Daniel A. Horner & C. William McCurdy Lawrence Berkeley National Laboratory, Berkeley, California

2. Contents • Construction of a complex B-spline basis set in exterior complex scaling calculations • Examples in one dimension • Photoionization of Helium • Photo double ionization of Helium • Photoionization of H2

3. Construction of B-spline basis sets

4. Example B-splines of order 8. Box size R=60 and 22 knot points

5. Im(r) q New recursive formulae: R0 Re(r) Exterior complex scaling (ECS) with B-splines Complex contour in ECS: B-splines are placed in the complex contour with one knot at r = R0.

6. Re(Bik) Im(Bik) Example B-splines of order 8 with R0 = 19.69 and Q = 40o Thin lines: real parts Thick lines: imaginary parts

7. These correctly impose the ECS condition: B-splines that do not straddle the point R0 are real, whether they are in the complex part of the contour or not ! Properties

8. Spectrum of a free particle m = 918.075; Rmax = 60; B-splines of order 8

9. Morse potential with centrifugal barrier

10. Morse potential with centrifugal barrier m = 918.075; Rmax = 60; B-splines of order 10; R0 = 40; q = 10o

11. He photoionization Driven Schrödinger equation for one-photon absorption: F(+)describes photo excitation as well as single and double ionization with purely outgoing wave boundary conditions Total photoionization cross section:

12. Complex Helium spectrum Im(E) Re(E)

13. Wavefunctions Contributions of the continua (ks,kp) and (kp,kd) to Re((+)) for h=20eV above the double ionization threshold Double ionization resides in the region of large r1 and r2 Single ionization resides in the regions where only one of r1 or r2 is large

14. He photoionization

15. x k1 f1 z q1 q2 Polarization axis k2 y Photo double ionization Double ionization amplitude is given by the finite-volume matrix element k1(-): Coulomb function with chargez=Znuc hi: single-electron Hamiltonian Triply differential cross section (TDCS)

16. Singly differential cross section Exp: Brauning et al, J. Phys. B 31 5149 (1998). TD-CC: J. Colgan et al, J. Phys. B 34 L457 (2001).

17. TDCS: Equal energy sharing Exp: Brauning et al, J. Phys. B 31 5149 (1998). Black line: kskp, kpkd and kdkf

18. TDCS: Unequal energy sharing and q1 = 0o Exp: Brauning et al, J. Phys. B 31 5149 (1998). TD-CC: J. Colgan et al, J. Phys. B 34 L457 (2001). HRM-SOW: P. Selles et al, Phys. Rev. A 65 032711 (2002).

19. TDCS: Unequal energy sharing and q1 = 30o Exp: Brauning et al, J. Phys. B 31 5149 (1998). Black line: kskp, kpkd and kdkf

20. TDCS: Unequal energy sharing and q1 = 60o Exp: Brauning et al, J. Phys. B 31 5149 (1998). Black line: kskp, kpkd and kdkf

21. TDCS: Unequal energy sharing and q1 = 90o Exp: Brauning et al, J. Phys. B 31 5149 (1998). Black line: kskp, kpkd and kdkf

22. Conclusions • ECS + B-spline basis sets: a powerful tool for the treatment of the double ionization continuum • Accurate results for SDCS and TDCS in photodouble ionization of Helium • Application to H2 in progress