J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 5585–5591. Printed in the UK PII: S0953-4075(00)16391-6 Charge capture and impact excitation processes in H + on He + collisions: a case study by the time-dependent Schr ¨ odinger equation method Xiao-Min Tong†, Daiji Kato†, Tsutomu Watanabe† and Shunsuke Ohtani†‡ † Cold Trapped Ions Project, ICORP, Japan Science and Technology Corporation (JST), Axis 3F, 1-40-2 Fuda Chofu, Tokyo 182-0024, Japan ‡ University of Electro-Communication, Chofu, Tokyo 182-0021, Japan E-mail: tong@hci.jst.go.jp Received 14 August 2000, in final form 29 September 2000 Abstract. We have studied the charge capture and impact excitation processes in H + on He + collisions over a wide range of collision energies by solving the time-dependent Schr¨ odinger equation with the classical trajectory approximation for the projectile. The time-dependent Schr¨ odinger equation is solved by the split-operator method with a generalized pseudospectral (non- uniform grid) method in the energy representation. The calculated charge capture cross sections are in good agreement with the available experimental measurements. Our calculated charge capture and impact excitation cross sections are also in reasonable agreement with various close-coupling calculations. Combined with time-dependent density functional theory, our Schr¨ odinger equation method (time propagation) holds significant promise for studying many-electron processes in atom– ion collisions. 1. Introduction Charge capture, impact excitation and ionization are the fundamental processes in atom collisions with atoms, molecules and solid material, and have been of considerable interest both theoretically and experimentally for a long time [1–3]. Due to recent advances in computer technology, we can solve the one-electron time-dependent Schr¨ odinger equation accurately to study various processes involved in the collisions [4,5]. Even with the largest computer, however, we are still far from solving the N -electron time-dependent Schr¨ odinger equation numerically. Recently, Nagano et al [6] studied the many-electron charge transfer process between highly charged ions and atoms using the time-dependent local density approximation method. Their results are not in good agreement with the experimental measurements. The discrepancies could be due to the time-dependent local density approximation or the numerical approach, which need further investigation. For one-electron systems, the close- coupling method shows great success in investigating the charge transfer, impact excitation and ionization in atom–ion collisions [1,3]. However, it is very difficult to extend the close- coupling method to study many-electron systems. The time-dependent density functional theory provides a way to study many-electron processes in the collision system. Basically, we only need to solve a single-electron-like time-dependent Schr¨ odinger equation to study many-electron processes. Therefore, it is very important to have an efficient and accurate time propagation method. The split-operator method with a generalized pseudospectral method 0953-4075/00/245585+07$30.00 © 2000 IOP Publishing Ltd 5585