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We formulate exact integral boundary conditions for a solution of the time-dependent Schrödinger equation that describes an atom interacting, in the dipole approximation, with a laser pulse. These conditions are imposed on a surface ͑boundary͒ which is usually chosen at a finite ͑but sufficiently remote͒ distance from the atom where the motion of electrons can be assumed to be semiclassical. For the numerical integration of the Schrödinger equation, these boundary conditions may be used to replace mask functions and diffuse absorbing potentials applied at the edge of the integration grid. These latter are usually introduced in order to ͑approxi-mately͒ compensate for unphysical reflection which occurs at the boundary of a finite region if a zero-value condition is imposed there on the solution. The present method allows one to reduce significantly the size of the space domain needed for numerical integration. Considering the numerical solution for a one-dimensional model, we demonstrate the effectiveness of our approach in comparison with some other numerical methods.

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The multifold differential cross section of the ionization of hydrogen molecular ion by fast-electron impact is calculated by a direct approach, which involves the reduction of the initial six-dimensional ͑6D͒ Schrödinger equation to a 3D evolution problem followed by the numerical modeling of the wave-packet dynamics. This approach avoids the use of stationary Coulomb two-center functions of the continuous spectrum of the ejected electron that demands cumbersome calculations. The results obtained, after verification of the procedure in the case of atomic hydrogen, reveal interesting mechanisms in the case of small scattering angles.

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