Nonadiabatic electron dynamics effects on high-harmonic generation spectrum of H$_2^+$: minima and oscillatory pattern

We numerically solved the full-dimensional electronic time-dependent Schr\"{o}dinger equation for H$_2^+$ with Born-Oppenheimer approximation under different sin$^2$-shaped and trapezoidal laser pulses at some different wavelengths, with $I=$1 $\times 10^{13}$ Wcm$^{-2}$, 3 $\times 10^{13}$ Wcm$^{-2}$, and 6 $\times 10^{13}$ Wcm$^{-2}$ intensity at 4.73 a.u. and 7.0 a.u. internuclear distances. Some structures such as complexity, minima, and oscillatory patterns appeared in the high-order harmonic generation (HHG) spectra are investigated in this work by considering the electron localization, electron nonadiabatic dynamics, spatially asymmetric of the HHG, and the Rabi frequency of the population of the ground and excited electronic states to better understand the origins of these structures in the HHG spectrum. We will clear that the origin of complicated patterns of the HHG spectra in sin$^2$-shaped laser pulse is due to that the most portion of the HHG emission occurs at the falling part of the laser pulse. We explore that the oscillatory pattern in the HHG spectra originate from an oscillatory pattern in the $S_g$ and $S_u$ spectra and these oscillatory patterns in turn are due to the nonadiabatic electronic behavior appeared as the slow oscillation pattern in the ground and first excited electronic states populations. Also, our result shows that the minima of the HHG are related to the oscillatory patterns in $S_g$ and $S_u$ spectra.