We have theoretically investigated the high harmonic generation (HHG) spectra of H$_{2}^{+}$ and HD⁺ using a time-dependent wave packet approach for the nuclear motion with pulsed lasers of peak intensities $(I_{0})$ of $3.5 \times 10^{14}$ and $4.5 \times 10^{14}$ W/cm², wavelengths ($\lambda_{L}$) of 800 and 1064 nm, and pulse durations (𝑇) of 40 and 50 fs, for initial vibrational levels $\nu_{0} = 0$ and 1. We have argued that for these conditions the harmonic generation due to the transitions in the electronic continuum by tunnelling or multiphoton ionization will not be important. Thus, the characteristic features of HHG spectra in our model arise only due to the nuclear motions on the two lowest field-coupled electronic states between which both interelectronic and intraelectronic (due to intrinsic dipole moments, for HD⁺) radiative transitions can take place. For HD⁺, the effect of nonadiabatic (NA) interaction between the two lowest Born–Oppenheimer (BO) electronic states has been taken into account and comparison has been made with the HHG spectra of HD⁺ obtained in the BO approximation. Even harmonics and a second plateau in the HHG spectra of HD⁺ with the NA interaction and hyper-Raman lines in the spectra of both H$_{2}^{+}$ and HD⁺ for $\nu_{0} = 1$ have been observed for higher value of $I_{0}$ or $\lambda_{L}$. Our calculations indicate reasonable efficiencies of harmonic generation even without involving the electronic continuum.

We have theoretically investigated the high harmonic generation (HHG) spectra of H$_2^+$ and HD$^+$ using a time-dependent wave packet approach for the nuclear motion with combined twocolour (1$\omega_L$–3$\omega_L$) pulsed lasers for ωL corresponding to wavelengths 1064 nm and 800 nm. The 1$\omega_L$ and 3$\omega_L$ lasers have peak intensities of $I_1^0 = 5.0 \times 10^{13}$ W/cm$^2$ and $I_2^0 = 2.0 \times 10^{14}$ W/cm$^2$, respectively. We have taken the pulse duration of $T = 50$ fs for both the fields, and the molecular initial vibrational level $v_0 = 0$. We have argued that for these combinations, the harmonic generation due to transitions in the electronic continuum by tunnelling or multiphoton ionization may be neglected and only the electronic transitions within the two lowest electronic states would be important. Thus, the characteristic features of HHG spectra in the two-colour field are determined, in our model, by the nuclear motions on the two lowest field-coupled electronic states between which interelectronic and intraelectronic (due to the intrinsic dipole moments in case of HD$^+$) radiative transitions can take place. We have studied the role of relative phase ($\varphi$) of the two fields on the HHG spectra of the molecular ions. In case of HD$^+$, the effect of nonadiabatic (NA) nonradiative interaction between the two lowest Born–Oppenheimer (BO) electronic states (1$s\sigma_g$, 2$p\sigma_u$) has been taken into account. Our calculations give realistic HHG spectra which are reasonably efficient and extended for both H$_2^+$ and HD$^+$ in the mixed two-colour field without involving the electronic continuum. The use of two-colour (1$\omega_L$–3$\omega_L$) field enables us to generate high harmonics beyond that achievable with a single 1$\omega_L$ or 3$\omega_L$ field of the corresponding intensity, frequency and pulse time.