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The selection rules of high harmonic generation: the symmetries of molecules and laser fields

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.