## 3 Discussion
### The Electric Dipole-Magnetic Dipole Polarizability and ECD  
Turning to chiral molecules and chiroptical spectroscopy, the \(\mathbf{\alpha}^{em}\) tensor becomes of relevance. To validate polar for this kind of calculations, we address in Figure 3 the case of hexahelicene. This is a representative molecule of the class of helicenes, which are \(\pi\)-conjugated systems consisting of ortho-fused benzene rings and have many possible applications [55]. They are characterized by inherent helical chirality, and possess interesting spectroscopic properties [56; 57; 58; 59; 60; 61]. Figure 2: The real and imaginary parts of the isotropic invariant of the electric dipole polarizability of 4,4\({}^{\prime}\)-diaminoazobenzene (\(\frac{1}{3}Tr(\mathbf{\alpha}^{ce})\)) computed by polar from a TDDFT calculation at PBE/cc-pVTZ level: (**a**) the chemical structure of 4,4\({}^{\prime}\)-diaminoazobenzene and the graphical representation of the computed equilibrium structure, which is essentially planar, with the -NH\({}_{2}\) groups slightly pyramidalized; and (**b**) the results obtained by the polar code for two choices of \(N\), demonstrating good convergence at \(N=100\) states in this photon energy range. To approach the conditions used in [54], the damping \(\Gamma\) for polar calculations is to 0.24 eV. As shown in Figure 3, the trace of the \(\mathbf{\alpha}^{em}\) polarizability tensor evaluated by polar on the basis of TDDFT calculations nicely match with the sign and relative magnitude of the rotatory strengths \(R_{0k}\) reported by [PERSON] within the same TDDFT calculation. In comparison with hexahelicene, we have considered the hexamer model of a helically coiled graphene nanoribbon of recent synthesis [62] (HGNR-6), which is also a \(\pi\)-conjugated system with inherent chirality (see Figure 4a). Interestingly, if we compare hexahelicene with HGNR-6 (for the same \(P\)-helicity), we obtain the same sign of the rotatory strength for the optically more active low energy \(\pi\rightarrow\pi^{*}\) transition (computed at 391 nm in HGNR-6, see Figure 4b). Notably, being \(\pi\)-conjugation more extended in HGNR-6 than in hexahelicene, this causes a significant red-shift of the wavelength of such transition, and higher values of its rotatory strength. This is evident by comparing Figures 3b and 4b. Furthermore, following the assignment of this low energy \(\pi\rightarrow\pi^{*}\) transition in hexahelicene to a helical sense-responsive feature (H-type band) [60], and by observing that the associated rotatory strengths have the same sign in \(P\)-hexahelicene and \(P\)-HGNR-6, we infer that such low-energy transition (computed at 391 nm) is of H-type also in HGNR-6. Figure 3: (**a**) Graphical representation of the equilibrium structure of \(P\)-hexahelicene (data from B3 LYP/6-31G(d,p) DFT calculation); and (**b**) comparison of the experimental UV-Vis absorption of hexahelicene and the electronic circular dichroism(ECD) spectra of \(P\)-hexahelicene and \(M\)-hexahelicene (from [60]) vs. the corresponding spectra simulated by polar (\(\Gamma\) = 0.5 eV) based on results from TDDFT calculations (200 states, B3 LYP/6-31G(d,p)). We have finally considered a third inherently chiral \(\pi\)-conjugated molecule, the higher fullerene C\({}_{84}\), which belongs to \(D_{2}\) point group symmetry [63] (see Figure 5). Similar to the case of another chiral fullerene, C\({}_{76}\)[64; 65; 66; 12; 67; 68], UV-Vis absorption and CD spectroscopies were among the techniques which were instrumental also in the discovery and characterization of C\({}_{84}\). Here, we limit ourselves to C\({}_{84}\), for which the assignment of the absolute configuration was carried out by means of DFT methods in reference [69].