# Presentation: 500757

## Electron scattering and ionization of helium with 3 active electrons in the S-wave model  

- To obtain an accurate ionization flux, we subtract the portion of the scattered wave function associated with the inelastic channels.
- The ionization contribution is computed via the flux through a hyper-surface as a function of energy sharing. This requires extrapolation, which is described below

**Exterior Complex Scaling**
- The ECS transformation of coordinates allows us to apply the correct asymptotic boundary conditions without the need for matching to an assumed form.

- Electron scattering and ionization of helium with 3 active electrons in the S-wave model

- Daniel A. Horner1, C. William McCurdy2, and Thomas N. Rescigno2
*1. Department of Chemistry, University of California, Berkeley, California 94720, USA*
*2. Computing Sciences, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA*

**Abstract**
- Results are presented for the application of exterior complex scaling to the problem of electron scattering and ionization of helium in the S-wave model. In our calculation, all electrons are treated equivalently and the outgoing, scattered, wave is computed through a time-dependent method in a discrete variable representation basis.

- Excitation cross sections for helium from the ground state, 11S, and the second excited state, 21S.
- Red points are the current ECS results, and black points are convergent close coupling results [5]. Where appropriate, filled symbols correspond to “s=0”, singlet excitation results, while open symbols correspond to “s=1”, excitation to triplet states.

**References**
- C. W. McCurdy, D. A. Horner, and T. N. Rescigno *Phys Rev A* **65**, 042714 (2002)
- C. W. McCurdy, D. A. Horner, and T. N. Rescigno *Phys Rev A* **63**, 022711 (2001)
- C. W. McCurdy and T. N. Rescigno *Phys Rev A* **62**, 032712 (2000)
- T. N. Rescigno and C. W. McCurdy *Phys Rev A* **62**, 032706 (2000)
- C. Plottke, I. Bray, D. V. Fursa, and A. T. Stelbovics  *Phys Rev A* **65**, 032701 (2002)

**Inelastic Cross Sections**
- The amplitude for excitation to state *i* is given by the matrix element
- Using Green’s Theorem, we convert the volume integral to the surface form which we compute.

- Real part of scattered wave function plotted as a function of (r1, r2, r3=2.0 au). Shown on the left is the result with 11S initial state and incident energy of 49eV. On the right is the result with 21S initial state and 14eV incident energy.

- The single differential, energy sharing, cross section is computed from the flux at =.
- Above, we show an example of how the flux is fit and then extrapolated to =.
- The panels to the right show the final, extrapolated, SDCS’s with 11S and 21S initial states.

**Time-dependent Approach**
- To obtain the scattered wave function, SC,we must solve the driven Schrödinger equation
- This is accomplished through a time dependent method
- The scattered wave function is then the Fourier Transform of a time evolved wave packet

**Calculation Parameters**
- The calculations used 153 DVR basis functions spread over 11 finite-elements of 15th order quadrature. The ECS angle was 300 with R0 as 101 au. Wave packets were evolved to a time of 500 atomic time units with a time step of 0.025.

**Discrete Variable Representation**
- The DVR basis functions, i, are defined using the points and weights (xi, wi) of Gauss-Lobatto quadrature.
- Matrix elements can be done via the underlying quadrature and local functions are diagonal in the DVR basis.

- Example of inelastic channel subtraction. Shown on the left is the real part of the scattered wave function before subtraction. On the right is the function after subtraction.