For a consistent and transparent understanding and interpretation of the origin and magnitude of ΔΕac, ΔΕzz as well as the factors that influence their size, it is important to realize that practically all these quantities are dominated by the influence of the ("many-body") Coulomb correlation energy combined with sublattice frustration, which gives rise to the staggered sublattice potential across the zigzag ends of finite length AGNRs. In fact, the sublattice frustration, which is the driving force for the generation of the end/edge states, as we have illustrated earlier, constitutes the largest (or even the full) contribution on the Coulomb correlation energy. The understanding that most (or all) of the Coulomb correlation energy is devoted to counterbalance the topological frustration between sublattice and molecular symmetry-groups is the starting (and key) point of the present investigation. This principle together with the established (hidden) strong contributions of aromaticity and shell structure constitute the basis for the deeper understanding of all these quantities (ΔΕac, ΔΕzz, Δεζζ, and ΔΕs). Thus, if we can properly alleviate the sublatticemolecular group symmetry frustration (which is equivalent with inversion/reflection symmetry conflict), under the natural constrains of shell structure and aromaticity, we could effectively account for the (largest part of) Coulomb correlation energy. At the same time this would explain why the open shell states (singlet or triplet) are not the real lower energy states, but rather "pseudo-states". 2.1 Calculation of ΔΕzz and ΔΕac from the one-body DFT calculations. Within the 1-electron approximation underlining the DFT and Hartree-Fock (HF) self-consistent fields, the symmetry frustration between molecular (D2h) and sublattice (C2V) symmetry groups can be alleviated by effectively breaking (or redefining) the symmetry of the additional degrees of freedom (besides spatial coordinates) i.e., the spin and/or pseudospin (for real-space calculations). In the first case we can introduce non-zero spin values preserving the molecular symmetry, whereas in the second case we are forced to break molecular symmetry, by introducing open-shell singlet states, which when optimized geometrically converge normally to C2v symmetric geometries compatible with sublattice symmetry, thus breaking the molecular symmetry as well. This occurs because the HOMO (and LUMO) orbitals of the open singlet are obtained, by construction, by mixing the HOMO and LUMO orbitals of the closed singlet. These orbitals have opposite parities, u, or g, (and opposite behaviour under σy or σxz reflection plane). Thus, by losing the σy (or σxz) reflection plane of the D2h symmetry group, as is illustrated in Fig. (a), we can get sublattice distribution with opposite sublattice points at the two ends (or antisymmetric with respect to y-axis, but symmetric with respect to the x-axis, which is the axis of the AGNR). This facilitates frontier molecular orbitals (HOMO, LUMO) localized only at one end (left or right) of the AGNR, producing an antisymmetric (pseudo)spin density, as is shown in Fig. , reflecting the sublattice symmetry and structure. Obviously, the reverse picture with the pseudospins interchanged is equally valid. On the other hand, the molecular D2h symmetry demands same type (same sublattice) atoms at the two ends, as shown in the lower part of Fig. (a), thus producing a symmetric, with respect to the y-axis, (pseudo)spin distribution (bottom of Fig. ). In both cases the (pseudo)spin distribution is almost zero at the middle part of the AGNR. This is reproduced in the corresponding "spin" densities (b). Such spin densities, in Figs. ), 1(c) and 1(d), which are generated self consistently (through the DFT convergence process) are in fact pseudospin densities. As mentioned above, the open singlet reflects the sublattice symmetry (with different type atoms at the two ends) with equal number of up and down (pseudo)spins, or A and B sublattice points, resulting in a balanced (nearly zero spin distribution in the middle). The triplet state on the other hand has also a region of zero spin in the middle, exactly where the sublattice imbalance occurs. In the ordinary "atomistic" calculations the sublattice degree of freedom does not enter in the spatial Hamiltonian and can only be introduced as (pseudo) spin. Then, due to the better account of Coulomb interaction, open shell states (triplet or singlet) appear energetically lower than the closed singlet state. This is because the additional degree of freedom of "pseudospin", introduced to take care of the sublattice topology (and the staggered potential), facilitates the optimization of Coulomb interaction by keeping away of each other electrons of different spin (for which Pauli repulsion is not operative), but of identical pseudospins. Moreover, based on the shell model , the unoccupied states of the "previous" (shell number smaller by 1) AGNR are the occupied of the current AGNR. This is responsible for the interplay between odd and even parity HOMOs as the width of AGNRs is growing (3n AGNRs have odd HOMO and even LUMO, whereas 3n+1 AGNRs are characterized by even HOMO and odd LUMO). FIGURE 1. Molecular and sublattice symmetry of the 3x6 (7,12) AGNR (a), reflected in the spin densities (b). Isovalue=0.004. The vertical elliptic curve indicates the region of sublattice imbalance (and frustration). Comparison of the corresponding spin densities for the 4x4 (9, 8) AGNR at the HF, MP2 and DFT/PBE0 level is given in (c) and (d). This is also responsible for the well-known 3n, 3n±1 width rule for AGNRs. Note that the (pseudo)spin densities invariably reflect the sublattice (pseudospin) structure within the frustrated molecular (D2h) symmetry 3 in the first case, or the sublattice symmetry (C2v) in the latter (see also Fig. ), where opposite end sites have opposite spins. It should be emphasized at this point that for wider AGNRs (where n>1 in the above rule for width 3 ), higher spin states are required 3 to lower the total energy (within the molecular D2h symmetry group). Such larger (pseudo) spin-polarized states optimize better the sublattice distribution (within the D2h molecular group), whereas the open-shell singlets lye higher in energy and revert to the closed singlet state. This illustrates emphatically that the open-singlet state is not the true ground (lowest energy) state of AGNRs (and, consequently, no conventional magnetism is truly present). Nevertheless, the open singlet state is still a very useful and efficient concept for the description of end-states, as is illustrated below. It should be emphasized that in both cases of Fig. , when correlation is introduced even at the MP2 level, the energetical ordering is reversed and the lowest energy structure is a closed singlet. In addition, the MP2 correlated "spin" density of the triplet, as we can see in Fig. , is rather correcting the HF failure (having the opposite sign) than reflecting the full sublattice structure. Note also in Fig. This should be a good estimate of the "missing" Coulomb energy in this case, and, based on the (approximate) electron-hole symmetry, the expected (HOMO-LUMO) separation of the open singlet (or the triplet) should be about twice as large (~2 eV). Indeed, the calculated open-singlet HOMO-LUMO gap for the 9-AGNRS (or 4x) is 2.2 eV, and so is ΔΕzz (vide infra). Even more important is the fact that the corresponding value for the 3x6 or (7,12) AGNR is also about 0.93 eV, suggesting an open-singlet gap of about 1.9 which is in excellent agreement with both the measured value of ΔΕzz (1.90 eV), and the calculated open singlet gap. It is important to observe also that the opensinglet value "ΔΕzz" = 1.2 eV for the 5-AGNRs and the 1.9 eV open singlet gap for the 7-AGNRs are practically equal to the correlation improved GW-LDA bandgap differences, which is highly suggestive for the essential correctness of our claim. Thus, within the one-electron approximation we have established the correct basis for discussion and analysis of both ΔEzz and ΔEac. ΔΕzz is identified as the open-singlet HOMO-LUMO gap, whereas ΔΕac can be identified as the difference |(HOMO-1)-(LUMO+1)|, with the understanding that both HOMO and LUMO are end-states. Nevertheless, as will be illustrated below, similar values of gaps and analogous estimates can be found in the triplet state as well. Furthermore, it should be emphasized that the central meaning of ΔΕzz is only valid for lengths L longer than the critical length ( L≥Lc), although the open-singlet HOMO-LUMO is defined for almost all lengths and is practically constant, as is verified also by Wang et al. 2.2 Introducing many-body corrections to the gaps. For both gaps (ΔΕac and ΔΕzz) we can further correct if we wish their (one-body) values by considering additional many body contributions through time-dependent DFT (TDDFT), which has been shown 1 to provide very good ("many-body") estimates of the gaps, so that the STS spectrum overall looks very much alike the (luminous) optical spectrum, because both are dominated by molecular overlaps between transition states. This is further illustrated and "verified" from the results below. Furthermore, the use of TDDFT allows the clear and unambiguous identification of the energy separation of the end/edge states, which according to the present investigation is not given by ΔΕzz, as Wang et al. have suggested, but by another type of gap which here is denoted as Δεζζ. In the usual one-body approximation Δεζζ corresponds to the HOMO-LUMO separation of the closed singlet true ground state for L≥Lc, which is always only a few 0.1 eV (~0.1 eV, for L→∞) in accord with the association of the end states with the Dirac points (and charge neutrality points 4 ) located "very close" to the fermi level. TDDFT indeed verifies that in contrast to Δεζζ which involves transition from one purely end-localized HOMO state to an opposite-parity end-localized LUMO state, the ΔEzz gap always involves transitions from a mixture (~ 60% -~ 40%) of "surface"-"bulk" states to another state of about equal amount of mixing. Thus, although ΔEzz involves a large amount of localized end-states, it should not be associated with the energy separation of the end-states. Another way, besides TDDFT, to distinguish between "bulk" and "surface" energy gaps is by comparing to the corresponding "edge-modified" AGNRs, obtained by eliminating "empty" (i.e., non-aromatic) end-rings, which also eliminates topological end-states (and, therefore, ΔEzz and Δεζζ).