## 3 Discussion
### Vibrational Polarizabilities and IR/VCD Spectroscopies  
To benchmark the vibrational contributions to molecular polarizabilities provided by polar, we begin by considering \(\mathbf{\alpha}^{ee,p}\) in the static limit. Such a quantity is straightforwardly provided by the Gaussian code in standard vibrational frequency calculations, and for small molecules (CH\({}_{4}\), CF\({}_{4}\), and CCl\({}_{4}\)) it is also available from published independent calculations [70] and experiments [71]. As shown in Table 1, the numerical data provided by polar have the same quality as those reported by Gaussian, and compare reasonably with other theoretical data and experimental results. Parallel to the case of the electronic polarizability, where the imaginary part of \(\mathbf{\alpha}^{ee}(-\omega;\omega)\) relates to UV-Vis absorption spectroscopy and the real part of \(\mathbf{\alpha}^{em}(-\omega;\omega)\) relates to circular dichroism (CD) spectroscopy, the calculation of the frequency-dependent vibrational polarizabilities \(\mathbf{\alpha}^{ee,p}(-\omega;\omega)\) and \(\mathbf{\alpha}^{em,p}(-\omega;\omega)\) paves the way to the simulation of IR and VCD spectra, respectively. It is known that VCD and IR (to a minor extent) are fairly sensitive to structural changes in flexible molecules [72; 73] and to the equilibrium among conformers [74; 75; 76; 77]. Even though it is possible to handle molecular flexibility in VCD (e.g., see Ref. [78] for a recent approach), in our validation of polar, we decided to consider rigid molecules, for the sake of simplicity. We selected four chiral molecules with unique conformations, which offers a more straightforward comparison with the experimental IR and VCD spectra. The application of polar to molecules with many conformations is a conceptually straightforward task, which could be considered in future versions of the code. The experimental VCD and IR spectra of (15)-Fenchone ((15) -FEN), (15)-Camphor ((15)-CAM), (15)-2-Methylenedone ((15)-MEFEN) and (15)-2-Methylenedmop ((15)-MECAM) were investigated in [79], and the main features of the spectra where assigned with the help of DFT calculations. The satisfactory comparison between frequency-dependent polarizabilities and experimental IR and VCD spectra of the four molecules is presented in Figures 6 and 7. The dissymmetry factor \(g\) plays an important role in VCD spectroscopy [80; 81]. The \(g\) factor analytically correlates with the concept of robustness [82; 83], which was introduced in VCD by [PERSON] and [PERSON] [84]. This useful parameter characterizes the soundness of the numerical prediction of VCD intensities and signs, and its use has been extended by [PERSON] and [PERSON] through the dissymmetry factor [85]. For these reasons, we have decided to compare in Figure 8 the function \(g(\omega)\) computed by polar with the corresponding experimental quantity measured for 15-CAM, taken as a representative reference. Both the order of magnitude and the spectral shape of the computed \(g(\omega)\) match reasonably well with the experimental results, which we take as a solid benchmark of the implementation of the polar code. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Molecule** & \(\frac{1}{3}\mathbf{Tr(\mathbf{\alpha}^{ee,p})}\) & **From Gaussian09 Output** & **From Ref. [70]** & **Expt. from Ref. [71]** \\ \hline CH\({}_{4}\) & 0.2371 & 0.2354 & 0.2299 & 0.2025 (0.2025) \\ CF\({}_{4}\) & 7.3981 & 7.3450 & 8.0347 & 5.3988 (7.6259) \\ CCl\({}_{4}\) & 8.1828 & 8.1241 & 6.3524 & 5.6688 (6.4111) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vibrational electric polarizabilities of CH\({}_{4}\), CF\({}_{4}\), and CCl\({}_{4}\) (data are in atomic units [41]; B3 LYP/aug-cc-pVTZ DFT calculations for Columns 1 and 2—present work). The minor discrepancy between Columns 1 and 2 is due to slight differences in the numerical values of the physical constants involved in the calculations (i.e., polar in Column 1 vs. Gaussian in Column 2).