Photoionization of atoms and molecules studied by the Crank-Nicolson method

The Crank-Nicolson (C-N) method combined with a B-spline basis set can be used in both time-dependent and time-independent calculations of the photoionization cross sections of atoms and molecules. For time-independent systems, the imaginary-time propagation (ITP) method can usually only converge an arbitrary initial state to the ground state directly. Contrary to existing methods, it is found that the C-N method can converge an arbitrary initial state directly to not only the ground state but also excited and continuum states by controlling the time-step size. It is very useful if one is interested in only part of the spectral information since the computation is relatively cheap. The C-N method can also be directly applied in time-dependent calculations. During the time evolution, the spectral information, such as energy and momentum, can be retrieved by projecting the wave function on the eigenstates obtained using the ITP method. Both time-dependent and time-independent calculations agree very well with previous results. This method can also be extended to two-electron systems directly.