The numerical approximation of the solution of the time-dependent Schrödinger equation
arising in ultrafast laser dynamics is discussed. The linear
Schrödinger equation is reduced to a computationally tractable,
lower dimensional system of nonlinear partial differential equations by the
multi-configuration time-dependent Hartree-Fock method. This method
serves to approximate the original wave function on a nonlinear manifold,
using the antisymmetry inherent in the model to significantly reduce the dimension
of the solution space. For the solution
of the resulting systems of PDEs, several numerical techniques are compared.
Space discretization using the pseudospectral method turns out to be superior
to finite difference approximations. For time integration, the range
of applicability and computational efficiency of high-order Runge-Kutta methods
are compared with variational splitting, a method recently proposed for
quantum molecular dynamics.