# Presentation: 789574

## 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

## Contents

**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

## Construction of B-spline basis sets

**Construction of B-spline basis sets**

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

**Example**

## Exterior complex scaling (ECS) with B-splines

- B-splines are placed in the complex contour with _one knot at __*r = R*__*0*_.

**Exterior complex scaling (ECS) with B-splines**

- Complex contour in ECS:

## B-splines of order 8 with R0 = 19.69 and = 40o 

- Thin lines: real parts

- Thick lines: imaginary parts

**Example**

## Properties

**Properties**

## Spectrum of a free particle

**Spectrum of a free particle**

**= 918.075; *R*max = 60; B-splines of order 8

## Morse potential with centrifugal barrier

**Morse potential with centrifugal barrier**

## Morse potential with centrifugal barrier

**Morse potential with centrifugal barrier**

**= 918.075; *R*max = 60; B-splines of order 10; *R*0 = 40; ** = 10o

## (+) describes photo excitation as well as single and double ionization 

**He photoionization**

- (+) describes photo excitation as well as single and double ionization
- with ***purely outgoing wave boundary conditions***

- Driven Schrödinger equation for one-photon absorption:

- Total photoionization cross section:

## Complex Helium spectrum

**Complex Helium spectrum**

- Im(E)

- Re(E)

## Wavefunctions

**Wavefunctions**

- Contributions of the continua (*k*s,*k*p) and (*k*p,*k*d) to Re((+)) for *h*=20eV above
- the double ionization threshold
- Double ionization resides in the region of large* r*1 and *r*2
- Single ionization resides in the regions where only one of *r*1 or *r*2 is large

## He photoionization

**He photoionization**

## k1(-): Coulomb function with charge z=Znuc

**Photo double ionization**

****k**1(-): Coulomb function with charge *z*=*Z*nuc
* **h**i*: single-electron Hamiltonian

**Double ionization amplitude **is given by the finite-volume matrix element

**Triply differential cross section (TDCS)**

## Singly differential cross section

**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).

## TDCS: Equal energy sharing

**TDCS: Equal energy sharing**

- Exp: Brauning et al, J. Phys. B **31** 5149 (1998).   Black line: kskp, kpkd and kdkf

## TDCS: Unequal energy sharing and 1 = 0o

**TDCS: Unequal energy sharing and ********1**** = 0****o**

- 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).

## TDCS: Unequal energy sharing and 1 = 30o

**TDCS: Unequal energy sharing and ********1**** = 30****o**

- Exp: Brauning et al, J. Phys. B **31** 5149 (1998). Black line: kskp, kpkd and kdkf

## TDCS: Unequal energy sharing and 1 = 60o

**TDCS: Unequal energy sharing and ********1**** = 60****o**

- Exp: Brauning et al, J. Phys. B **31** 5149 (1998). Black line: kskp, kpkd and kdkf

## TDCS: Unequal energy sharing and 1 = 90o

**TDCS: Unequal energy sharing and ********1**** = 90****o**

- Exp: Brauning et al, J. Phys. B **31** 5149 (1998). Black line: kskp, kpkd and kdkf

## Conclusions

**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