  We present theoretical calculations of the Land\'e g-factors of semiconductor
nanostructures using a time-dependent empirical tight-binding method. The
eigenenergies and eigenfunctions of the band edge states are calculated as a
function of an external magnetic field with the electromagnetic field
incorporated into the tight-binding Hamiltonian in a gauge-invariant form. The
spin-orbit interaction and magnetic field are treated non-perturbatively. The
g-factors are extracted from the energy splitting of the eigenstates induced by
the applied magnetic field. Both electron and hole g-factors are investigated
for CdSe nanostructures. The size and aspect ratio dependence of g-factors is
studied. We observe that the electron g-factors are anisotropic and find that
the calculated values agree quantitatively with experimental data. We conclude
that the two distinct g-factor values extracted from time resolved Faraday
rotation experiments should be assigned to the anisotropic in plane
($g_\parallel$) and out of plane ($g_z$) electron g-factors, rather than to
isotropic electron and exciton g-factors. We find that the anisotropy in the
electron g-factor depends on the aspect ratio of the nanocrystal. The g-factor
anisotropy derived from the wurtzite structure and from the non-unity aspect
ratio may cancel each other in some regime. We observe that hole g-factors
oscillate as a function of size, due to size dependent mixing between the heavy
hole-light hole components of the valence band edge states. Extension to the
calculation of exciton g-factors is discussed.
