**
Hyperspherical Approach for Ion-Atom Collisions at Low Energies
**

Teck G. Lee, B.D. Esry and C. D. Lin

Department of Physics, Kansas State University, Manhattan, Kansas 66506, U.S.A.

Single electron transfer and excitation processes in slow collisions of singly/multiply charged ions with hydrogenic atoms/ions is one of the most basic reactions in a three-body Coulomb system. For such collisions, a variety of quantum mechanical approaches have been proposed in the past. One of the well-developed approaches is the hyperspherical close-coupling method. This method has been used extensively for the collisions of electrons and positrons with atomic hydrogen as well as collisions involving muons [1]. The method, however, has never been applied to ion-atom collisions until recently [2,3].

In references [2,3], the two-channel adiabatic hyperspherical method has been applied to calculate the elastic and charge transfer reactions in simple collision systems at low energies. To extend the method to excitation and charge transfer to excited states, however, one must include higher channels. To avoid treating the sharp avoided crossings in the adiabatic approach, the so-called 'diabatic-by-sector' method has been employed previously. However, this method has its limitations [4]. An alternative hyperspherical formulation [5] has been developed, also within the spirit of the diabatic approach. This method is adopted here for treating ion-atom collisions at low energies.

Consider a system described by variables (R,Ω), where R is the hyperradius and Ω denotes all the other angular coordinates. Starting with the time independent Schrodinger equation in the body-frame

(1)

where T(R) is the kinetic energy operator for motion in R and H_{ad} is the adiabatic
Hamiltonian in Ω which depends parametrically on R. We seek solutions of (1) in the
form

(2)

where is a B-spline basis function and denotes the channel function. The index j represents the radial grid, I is the absolute value of the projection of total angular momentum J along the body frame quantization axis, and v denotes the adiabatic channel number. Substituting (2) into (1), we obtain a generalized algebraic eigenvalue problem for the coefficients .

The method has been tested for bound states. For instance, the HD+ system,
with 1 channel calculation, we obtained the binding energy of the lowest state
at 21515.056 cm^{-1}, which is to be compared with the exact value of
21515.981 cm^{-1}.
In combination with the R-matrix propagation technique, this method is being
extended to scattering calculations to obtain cross sections for excitation and
charge transfer in ion-atom collisions at low energies.

**References:**

1) C.D. Lin, Phys. Rept. 257, 1 (1995)

2) A. Igarashi and C.D. Lin, Phys. Rev. Lett. 83, 4041 (1999)

3) Z.X. Zhao, A. Igarashi and C.D. Lin, Phys. Rev. A. 62, 042706 (2000)

4) Ken-ichi Hino, Akinori Igarashi and Joseph H. Macek, Phys. Rev. A. 56 1038 (1997)

5) B. D. Esry, Chris H. Greene and James P. Burke Jr. (unpublished)

This work was supported by the
Chemical Sciences, Geosciences and Biosciences Division,

Office of Basic Energy Sciences,
Office of Science,
U.S. Department of Energy.

*Submitted to ICPEAC 2001, July 2001 in Santa Fe, NM.*

*This abstract is also available in
Postscript or
Adobe Acrobat formats.*