Atomic and Molecular Double Photoionization Calculations Using Exterior Complex Scaling

1. Atomic and Molecular Double Photoionization Calculations Using Exterior Complex Scaling C. William McCurdy, Dan Horner, Tom Rescigno Lawrence Berkeley National Laboratory, Berkeley, California And Fernando Martín Departamento de Química, Universidad Autónoma de Madrid Spain Funded by DOE Basic Energy Sciences

2. Introduction and Outline • Double photionization requires coulomb boundary conditions for two outgoing electrons • Exterior complex scaling (ECS) allows the application of those boundary conditions without their explicit imposition • Has been used to solve the electron-impact ionization problem [Science ...] • B-splines provide a new representation of the Hamiltonian under exterior complex scaling – • A molecular single-center expansion implementation of B-splines already exists (F. Martín and coworkers) • Construction of a complex B-spline basis set for exterior complex scaling calculations • Complete solution for photo double ionization of Helium • First steps for molecules: Photoionization of H2

3. Im(r) q R0 Re(r) Exterior complex scaling (ECS) and Boundary Conditions For photoabsorption we must solve a driven Schrodinger equation With purely outgoing boundary conditions on F(+) for both electrons Scaling the radial coordinates of both electrons applies the +ih boundary condition and produces pure outgoing waves

4. Implementation with B-splines • Implementation requires a “grid” like basis e.g. finite difference, finite element or discrete variable representation • For ECS B-splines are another choice with knots placed on the complex contour with one knot at r = R0. • New recursive formula for B-splines under ECS

5. B-splines of order 8 with R0 = 19.69 and Q = 40o Re(Bik) Im(Bik) These correctly impose the ECS condition: Properties of Complex B-Splines Im(r) q R0 Re(r) B-splines that do not straddle the point R0 are real, whether they are in the complex part of the contour or not !

6. Wavefunctions (1S 1P transitions) Contributions of the continua (ks,kp) and (kp,kd) to Re((+)) for h=20eV above the double ionization threshold (kp,kd) (ks,kp) 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

7. 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)

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

9. 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).

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

11. Polarization direction k1 TDCS: Equal energy sharing 20 eV excess photon energy

12. H2 Single Photoionization – Parallel polarization Fixed-nuclei single photoionization cross section for H2 by exterior complex scaling in B-splines Present results I. Sanchez and F. Martin J. Phys B 30 679 (1997)

13. 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 including the construction of H2+ states for evaluating the amplitude for double ionization