  The selection rules of high harmonic generation (HHG) are investigated using
three-dimensional time-dependent density functional theory (TDDFT). From the
harmonic spectra obtained with various real molecules and different forms of
laser fields, several factors that contribute to selection rules are revealed.
Extending the targets to stereoscopic molecules, it is shown that the allowed
harmonics are dependent on the symmetries of the projections of molecules. For
laser fields, the symmetries contributing to the selection rules are discussed
according to Lissajous figures and their dynamical directivities. All the
phenomena are explained by the symmetry of the full time-dependent Hamiltonian
under a combined transformation. We present a systematic study on the selection
rules and propose an intuitive method for the judgment of allowed harmonic
orders, which can be extended to more complex molecules and various forms of
laser pulses.
