  High-order harmonic generation (HHG) from molecular ion HeH$^{2+}$ in two initial states of $1s\sigma$ (ground state) and $2p\sigma$ (first excited state) is investigated theoretically, in homogeneous and plasmonic-enhanced laser fields. The electron ionization and the enhancement of HHG yield in ground and first excited states, as the initial states, is studied and a strong orientation effect for ionization of electron in $2p\sigma$ state is discovered, that it can be used to reach a longer lifetime for the electron in the first excited state, which is responsible for the enhancement in HHG yield, during the high harmonic emission. In order to investigate effect of plasmonic field on the cutoff position of HHG spectra, first the enhancement of laser field in a nano-antennae is calculated and then the interaction of the enhanced field with HeH$^{2+}$ molecule is studied by solving the time-dependent Schr\"{o}dinger equation. According to the results, when HeH$^{2+}$ molecule, in the initial state of $2p\sigma$, is irradiated by plasmonic polarized laser field, with the polarization normal to the molecular axis, the HHG yield and cutoff position enhance while the excited state has a long lifetime during the HHG process. 
Quantum mechanical study of high-order harmonic generation from   HeH$^{2+}$ molecule in homogeneous and plasmonic-enhanced laser Fields