## 3 Results and Discussions
### Graphene Molecules without/with Hydrogen Passivation  
The optical absorption spectra for the graphene molecules without and with passivation are shown in Figure 1. We first check the graphene molecule without hydrogen passivation. We notice that the first absorption peak is at \(\sim 0.3\) eV. The absorption of such low energy indicates that the gap between LUMO and HOMO is small. We plot the eigenstates of the molecules in Figure 2. The energy gap of \(\sim\)0.197 eV is found from the energy spectrum. The first absorption peak (denoted in Figure 1(a)) corresponds to the electron excitation from the HOMO-1 to the LUMO, rather than HOMO-LUMO excitation. This is because the HOMO is formed by the \(p_{z}\) orbital (\(\pi\)-electron), while the LUMO is composed with the hybridized \(sp^{2}\) orbital (\(\sigma\)-electron). That is to say, the optical transition between HOMO and LUMO is forbidden due to symmetry. We carefully choose the pulse photon energy to be \(\sim 0.042\) eV, which is approximately one seventh of the transition energy from the HOMO-1 to LUMO. The results with a laser amplitude of 0.2 V/A are shown in Figure 3(a). Harmonic generations up to the 19 th order are identified. The plateau of the HHG intensity is observed. There is a peak with the order number of \(\sim\)7.8, denoted with the empty square in Figure 3(a). This peak corresponds to the electron excitation from HOMO-1 to LUMO, which we attribute to the intrinsic emission, rather than the HHG (see discussions in Section 3.2). Sub-peaks near the major harmonic peaks are also identified (denoted by triangles in Figure 3(a)). The positions of these sub-peaks are not exactly at the integer orders, however the neighboring orders differ by one. In our case, each integer major peak with order of \(n\) has one sub-peak at the order of \((n-0.59375)\). The existence of these peaks is interesting and consistent with the previous works on benzene and other ring-shaped molecules [25,26]. The shifted peaks may originate from the limited duration of the laser pulse [27], which will be further discussed in Section 3.2. Figure 3: HHG spectra for the non-passivated (**a**) and passivated (**b**) graphene molecules. The parameters for the laser field are: \(\hbar\omega=0.042\) eV, \(A=0.2\) V/Å. The intensity of the passivated graphene is much smaller than that of the non-passivated one. The integer-order harmonics are denoted by circles, while the shifted peaks are denoted by triangles. The empty square denotes the intrinsic emission for HOMO-1 to LUMO. If we select a laser pulse with a higher frequency, the total number of HHG is reduced. For instance, for \(\omega=0.28\) eV the highest harmonic order is only 5 th. A larger laser amplitude is required for the other higher order peaks to occur. This is governed by the law for the harmonic cutoff of \(I_{p}+3.17U_{p}\)[28; 29; 30], where \(I_{p}\) is the ionization potential and \(U_{p}\) is the ponderomotive energy that \(U_{p}=E^{2}/4\omega^{2}\). Since the laser frequency is higher, the laser amplitude should be increased accordingly to obtain the same HHG order. When the dangling bonds are saturated by hydrogen atoms, the excessive electrons at the edge form the C\(-\)H covalent bonds. The electrons in the system become localized. This is revealed by the narrow absorption peaks in Figure 1(b). The LUMO-HOMO gap is 1.53 eV, much larger than that of the one without passivation. The HHG spectrum is plotted in Figure 3(b). Above all, the harmonic intensity is highly suppressed compared with the non-passivated one, and a reduction of more than two orders is found. This shows that the unpaired electrons on the edge of the non-passivated graphene enhance the non-linear optical response due to the weak localization [31]. Furthermore, the total number of higher harmonics is also suppressed. Only the 3 rd order peak can be identified with the same laser pulse parameter (\(\omega=0.042\) eV, \(A=0.2\) V/A). This reveals that the ionization energy for this structure is much higher than the non-passivated one.