## 3 Red Shift of HHG in Resonant Systems
### Orientation Dependence of Red Shift in MHOHG  
From the above discussion, the resonance of \(2\,p\sigma\) state with long life time plays a key role in the red shift of MHOHG. The radiative transition from the ground \(1s\sigma\) state to the resonant \(2\,p\sigma\) state is however restricted to the parallel component of the laser field. In this subsection, we explore the orientation dependence of the red shift in MHOHG [22]. The numerical results of MHOHG spectra with different orientation angles \(\chi\) and the corresponding time profiles are presented in Figure 6a,b, respectively.  
An obvious feature in Figure 6a is that the overall intensity of MHOHG is strongly dependent on the resonance of the excited state \(2\,p\sigma\). The intensity of MHOHG in the plateau with orientation angle \(\chi\)=0' (parallel) is more than two orders higher than that with orientation angle \(\chi\)=90\(\lx@math@degree\) (perpendicular). The reason is that the direct ionization rate from the ground state \(1s\sigma\) is very low due to the large ionization potential. However, the EE to the resonant excited state \(2\,p\sigma\) will lead to EI, which greatly increases the ionization rate of the system. For the parallel orientation, the amplitude of radiative transition to the \(2\,p\sigma\) state is maximal. This corresponds to an efficient laser-induced electron transfer (LIET) in a nonsymmetric system [20]. One can observe a strong resonance in the MHOHG spectra. When we gradually increase the orientation angle \(\chi\) between the molecular axis and laser polarization,  
Figure 5: **(a)** Red shifted harmonics in the plateau with order \(N\) = 40–42 at different laser pulse durations \(\tau\). With increasing the pulse duration, the nonadiabatic effect is reducing and the redshift of the harmonics becomes smaller. **(b)** The corresponding time profiles of harmonic order \(N\) = 40. When the laser pulse duration is increased, more harmonics generated with \(t\) < 0, and the asymmetry between \(t\) < 0 and \(t\) > 0 is less, which reduces the red shift of the harmonics. The curves have been shifted vertically for clarity.  
the transition amplitude along the molecular axis direction diminishes, and the intensity of the resonance and the plateau in MHOHG decreases. It can be observed in the corresponding time profiles in Figure 6b. For perpendicular orientation with \(\chi\)=90 deg, the transition from the 1s\(\sigma\) state to 2\(\rho\sigma\) state is forbidden by selection rules. The electron may transit from the 1s\(\sigma\) state to 2\(\rho\pi\) state (\(-\)0.7388 a.u. (20.1 eV)). However, the 2\(\rho\pi\) state is not a resonant state with long life time. Its role in the MHOHG is therefore negligible. As a consequence, the above four-step model gradually changes to a three-step model with increasing angle \(\chi\), and the time profiles of higher-order harmonics illustrated in Figure 6b becomes more symmetric with respect to the time when the laser pulse reaches its peak intensity, _i.e._, \(t\) = 0. The red shift of MHOHG decreases with the increase of the orientation angle \(\chi\). No noticeable red shift occurs in the MHOHG spectra when \(\chi\)=90 deg as illustrated in the calculations presented in Figure 6a for the intensities and in Figure 6b for the time profiles. One may expect only odd-order harmonic generation due to the symmetry of the system with \(\chi\)=90 deg. However, a little population on the 2\(\rho\pi\) state will break the symmetry of the system, which will lead to even-order harmonic generation [22].  
Figure 6: (**a**) Orientation dependence of MHOHG in linearly polarized laser fields. (**b**) The corresponding time profile analysis. With increasing orientation angle, the intensity of the resonance in MHOHG around order 18 becomes weaker and weaker. The redshift in the inset of (**a**) is becoming less noticeable. For higher order MHOHG (larger than 40), the time-profile analysis in (**b**) shows increasing symmetry with increasing orientation angle.