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High-order harmonic generation (HHG) from molecular ion HeH$^{2+}$ in twoinitial states of $1s\sigma$ (ground state) and $2p\sigma$ (first excitedstate) is investigated theoretically, in homogeneous and plasmonic-enhancedlaser fields. The electron ionization and the enhancement of HHG yield inground and first excited states, as the initial states, is studied and a strongorientation effect for ionization of electron in $2p\sigma$ state isdiscovered, that it can be used to reach a longer lifetime for the electron inthe first excited state, which is responsible for the enhancement in HHG yield,during the high harmonic emission. In order to investigate effect of plasmonicfield on the cutoff position of HHG spectra, first the enhancement of laserfield in a nano-antennae is calculated and then the interaction of the enhancedfield with HeH$^{2+}$ molecule is studied by solving the time-dependentSchr\"{o}dinger equation. According to the results, when HeH$^{2+}$ molecule,in the initial state of $2p\sigma$, is irradiated by plasmonic polarized laserfield, with the polarization normal to the molecular axis, the HHG yield andcutoff position enhance while the excited state has a long lifetime during theHHG process.

Quantum mechanical study of high-order harmonic generation from HeH$^{2+}$ molecule in homogeneous and plasmonic-enhanced laser Fields