## 2 Results
### Advanced Frontiers Molecular Orbitals (FMOs)  
The frontier molecular orbitals, specifically the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), are key indicators of a molecule's electronic properties in quantum chemistry [43]. The HOMO represents electron donors, while the LUMO corresponds to electron acceptors, with light absorption being associated with the transition from HOMO to LUMO orbitals. Figure 4 shows the surface visualizations of the frontier molecular orbitals for the apigenin molecule. The HOMO orbitals are distributed across the entire molecule except for the hydrogen atoms, and the LUMO orbitals are similarly distributed, excluding hydrogen atoms and the carbonyl oxygen. The contour surfaces of the HOMO are mainly located over the \(\pi\)-system ring and the C7=C8 bond, where the accumulation of electron clouds on the oxygen atoms suggests a role in charge transfer properties. This distribution suggests a significant role of these atoms in charge transfer properties and interactions with biological targets. Notably, the C7=C8 bond and adjacent oxygen atoms facilitate electron delocalization, enhancing the molecule's reactivity in biological systems (Figure 5). These features are particularly important in modulating oxidative stress pathways, such as the \(\mathrm{Keap}\mathrm{1}-\mathrm{Nrf}\mathrm{2}\) interaction, by enabling stable yet flexible binding to the target protein. Figure 4: Surface visualizations depicting the frontier molecular orbitals of the apigenin molecule. Figure 5: Chemical structure of apigenin with numbered carbon positions. The \(\pi\)-system also contributes to the stabilization of \(\pi-\pi\) interactions with aromatic residues in proteins, highlighting apigenin's potential in receptor binding. Furthermore, the extensive HOMO delocalization reflects apigenin's strong antioxidant activity, as it allows for efficient electron donation to neutralize reactive oxygen species. The accumulation of electron clouds near these functional groups highlights their potential for facilitating binding and activity, particularly in pathways such as the Kaep1\(-\)Nrf2 interaction. Providing a detailed chemical structure with atom numbering would further enhance the interpretation of these findings, clarifying the specific roles of key molecular features. LUMO densities, on the other hand, are found primarily on the carbon atoms of the phenyl and phenol rings and on the central chain's C-C single bond. Thus, the electronic transition from HOMO to LUMO primarily arises from \(\pi-\)\(\pi^{\ast}\) electronic transitions due to the rich electron clouds of \(\pi\) atomic orbitals [44]. This transition underscores the molecule's ability to engage in light absorption processes, which may influence its bioavailability and interaction dynamics with proteins. The correlation between the calculated molecular energies and their quantum chemical parameters provides insights into the molecule's chemical reactivity. Quantum chemical descriptors, such as hardness (n), softness (S), electronegativity (x), and electrophilicity (\(\omega\)), serve as global indicators of a molecule's chemical behavior. The hardness value reflects the molecule's resistance to charge transfer, while softness measures the ability to accept electrons. Electronegativity (x) describes a molecule's tendency to attract electrons, and electrophilicity (\(\omega\)) determines the molecule's capacity to accept electrons [45]. Table 6 shows the calculated quantum parameters, highlighting that the studied molecule has a hardness value of 4.42 eV, a softness of 0.226 eV-1, and an electronegativity of 4.33 eV. The electrophilicity index (\(\omega\)) is 2.12 eV, classifying the apigenin molecule as a moderate electrophophile.