## 4 Materials and Methods  
All the polarizability calculations reported here were carried out through the polar code developed in this work. polar allows the numerical evaluation of the three polarizability tensors \(\mathbf{\alpha}^{ee}\), \(\mathbf{\alpha}^{em}\), \(\mathbf{\alpha}^{mm}\) as a function of the photon frequency \(\omega\). Both electronic and vibrational polarizabilities can be computed. polar is written in the C++ language, and adopts complex numbers for better code readability. Electronic polarizabilities are evaluated as sum over vertical electronic transitions (Equations (6)-(8)). The required input data are the excitation energies \(\hbar\omega_{k}\) of a selected set of excited states \(\ket{k}\), and the associated electric (magnetic) transition dipoles \(\mathbf{\mu}_{0k}\) (\(\mathbf{m}_{0k}\)). The output file of an ordinary TDDFT calculation made by Gaussian code [5] can be parsed by polar to gather all information required to operate. By writing additional parsers, polar can be straightforwardly interfaced to other quantum chemistry packages. The calculation of vibrational polarizabilities by Equation (13) requires the quantities computed by [PERSON] in a standard VCD job. These are the Hessian of the energy vs. the Cartesian nuclear displacements, and the derivatives of the electric and magnetic dipole vs. the Cartesian nuclear displacements (the latter two quantities are also, respectively, named APTs and AATs, see for instance [49; 50; 51]). By using the Hessian matrix, polar solves the secular equation to obtain the vibrational frequencies (\(\omega_{k}\)) and the normal modes, which are required to compute the dipole derivatives vs. normal coordinates (\(q_{k}\)) from APTs and AATs. The Hessian, APTs, and AATs are stored by [PERSON] in the formatted checkpoint file (\(\mathbf{\ast}\).fchk) and are read by polar before addressing the evaluation of vibrational polarizabilities. For convenience, since Gaussian checkpoint and output files report computed data in atomic units, we adopt the same convention in polar. The quite useful and neat paper by [PERSON] [41] can help the reader in converting all the quantities reported here from atomic to SI units. We merely remind here the following identities (where \(E_{0}=e^{2}/(\kappa_{0}a_{0})\) is the symbol for the Hartree, \(a_{0}\) the symbol for the Bohr, and \(\kappa_{0}=4\pi e_{0}\) is the electric permittivity):  
\[\left[\mathbf{\alpha}^{ee}\right]=\frac{\left[\mu\right]^{2}}{E_{0}}=\frac{\left(ea _{0}\right)^{2}}{\frac{e^{2}}{\kappa_{0}a_{0}}}=\kappa_{0}a_{0}^{3}, \tag{21}\]  
\[\left[\mathbf{\alpha}^{em}\right]=\frac{\left[\mu\right]\left[m\right]}{E_{0}}= \frac{\left(ea_{0}\right)\left(E_{0}ea_{0}^{2}/\hbar\right)}{E_{0}}=\frac{e^{ 2}}{\hbar}a_{0}^{3}, \tag{22}\]  
\[\left[\mathbf{\alpha}^{mm}\right]=\frac{\left[m\right]\left[m\right]}{E_{0}}= \frac{\left(E_{0}ea_{0}^{2}/\hbar\right)\left(E_{0}ea_{0}^{2}/\hbar\right)}{E _{0}}=\frac{E_{0}e^{2}a_{0}^{4}}{\hbar^{2}}. \tag{23}\]  
Since the atomic units of the electric and magnetic fields are \(\left[\mathcal{E}\right]=E_{0}/\left(ea_{0}\right)\) and \(\left[\mathcal{B}\right]=\hbar/\left(ea_{0}^{2}\right)\)[41], it is straightforward to show that \(\left[\mathcal{E}\mathbf{\alpha}^{ee}\mathcal{E}\right]=\left[\mathcal{E}\mathbf{ \alpha}^{em}\mathcal{B}\right]=\left[\mathcal{B}\mathbf{\alpha}^{mm}\mathcal{B} \right]=E_{0}\). The DFT and time-dependent density functional theory(TDDFT) calculations reported in this work have been carried out with Gaussian09 Rev. D.01 [5]. Because of the scattered nature of the literature reference data employed in this work to validate polar, we had to chose individual functionals and