  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.
