eV. Such conclusion is clearly considered here as highly improbable, in view of equally good (in fact better) agreement for the 7-and 9-AGNRs, not to mention Occam's principle. Moreover, if this is indeed a general trend, it clearly illustrates that elaborate correlation calculations (e.g., GW) could be avoided (see also ref. 23) if topological frustration can be taken into account appropriately by simple DFT (one particle) calculations, provided that the DFT functionals include "exact" exchange which is sensitive to inversion symmetry conflict. 3.2 7-AGNRs. Figure summarizes the results for the three spin states (closed singlet, open singlet, and triplet) for the 7-AGNRs, and in particular the (7, 12) or 3x6 AGNR. First of all, we can comment on the significance of the exact exchange in the DFT functional, which was discussed above. The calculated DFT/PBE0 open singlet ΔΕzz gap is 1.9 eV in full agreement with the measured ΔΕzz gap for the 3x6 (7, 12) AGNR. In contrast the ΔΕzz gap calculated with the PBE functional, which does not include "exact exchange", is less than half this value (~0.5 eV, in agreement with the PBE calculations of Wang et al. ). As we can see in Figs. there are gaps in all of them between HOMO (or HOMO-1) and LUMO, which are equal or very nearly equal to the measured ΔΕzz value of 1.9 eV. We must rremember also that this value is practically equal to the correlation energy obtained from the difference between the GW and LDA values 14 for these AGNRs. Let us first focus on the open singlet, which is commonly accepted as the "ground state". where the definitions of ΔΕzz and ΔΕac (which are designated as Δzz and Δac respectively) are illustrated. Moreover, the calculated DFT/PBE0 ΔΕzz and ΔΕac values (contrary to those of DFT/PBE, with no "exact exchange", used by Wang et al. ) are practically identical to the measured values for the 3x6 (7,12) AGNR deposited on non-metallic NaCl substrate (in full analogy to similar results for the 5-AGNRs, described above). Based on the closed singlet ground state, it becomes clear that the "real" energy separation of the end states is the HOMO-LUMO gap of the closed singlet state which is (almost always) about 0.1-0.3 eV (depending on the length). This is corroborated by the TDDFT results, giving rise to the Δεζζ gap, which discussed earlier for the 5-AGNRs, and is consistent with the appearance of Dirac points close and around the Fermi level, whereas ΔEzz is due to mixed transitions involving both "end" and "bulk" states. This is verified by Fig. ) which shows the excitation spectrum of the closed singlet state for the normal 3x6 (7,12) AGNR, in which there are two characteristic maxima at 1.9 eV and 3.2 eV, which practically coincide with the measured ΔEzz, and ΔEac values respectively for this AGNR. As we can see in the left part of Fig. 3(d), ΔEac involves transitions between (mixtures of) "bulk states" (from HOMO-1 and HOMO-2, to LUMO+1 and LUMO+2), whereas ΔEzz corresponds to transitions from mixt , "bulk" + "surface" (HOMO-3 and HOMO) to LUMO+3 and LUMO. Thus, ΔEzz, although not equal to energy separation of the end states, is clearly associated with the first (lowest energy) transition involving end and bulk states, corresponding to the measured ΔEzz value of 1.9