## 3 Discussion
### Introducing Intensity-Carrying Modes in VCD
However, the result of the summation in Equation (19) cannot be distinguished from the symmetric version below, which is preferred for numerical convenience:  
\[R_{0_{a}1_{a}}=\frac{1}{2}\sum_{ij}L_{ia}\mathcal{M}_{ij}^{VCD}L_{ja}, \tag{20}\]  
with  
\[\mathcal{M}_{ij}^{VCD}=\frac{\hbar}{2}\left[\frac{\partial\mathbf{\mu}}{\partial x _{i}}\cdot\frac{\partial\mathbf{\mu}}{\partial x_{j}}+\frac{\partial\mathbf{\mu}}{ \partial x_{j}}\cdot\frac{\partial\mathbf{\mu}}{\partial x_{i}}\right].\]From the inspection of Equation (20), we conclude that ICMs suitable for VCD can be found by seeking the stationary values of the rotatory strength \(R_{01}\) as a function of the nuclear displacement coefficients (subject to normalization of the vector of coefficients). Similar to the case of IR spectroscopy [33], this leads to the eigenvalue problem for the \(\mathbf{\mathcal{M}}\) matrix defined by Equation (20). We implemented the calculation of ICM in polar, for both IR and VCD spectroscopy, considering the \(\mathbf{\mathcal{M}}\) matrices given by Equations (18) and (20). The eigenvalues resulting from the solution of Equation (17) are collected in Table 3 (for IR) and in Table 4 (for VCD). We observe that the number of non-zero eigenvalues of \(\mathbf{\mathcal{M}}^{IR}\) is three (as previously found by [PERSON] [33]), while it is six for \(\mathbf{\mathcal{M}}^{VCD}\). As pointed out by [PERSON] et al. in discussing Raman and ROA ICMs [86], the rank of the \(\mathbf{\mathcal{M}}\) matrix indeed equals the number of independent components of the polarizability. Similarly, in VCD, we have three independent components of the electric and magnetic dipoles, which justifies the rank of \(\mathbf{\mathcal{M}}^{VCD}\) being six. Furthermore, since the sum of the rotatory strengths is zero [47] (i.e., \(\sum_{a}R_{0_{a}1_{a}}=0\)), this justifies the fact that the sum of the eigenvalues is approximately zero, within the numerical accuracy affordable by the selected computational method. The inspection of the data reported in Table 3 fully justifies the observation made in Ref. [79] that the IR intensities of FEN and CAM are overall stronger than MEFEN and MECAM (respectively). The eigenvalues of the IR-ICMs of the carbonyl-containing compounds FEN and CAM are quite similar, but they are remarkably larger than those of MEFEN and MECAM. Moreover, the inspection of the animations of the associated eigenmodes (Supplementary Materials) reveals that for \(\lambda_{1}\) and \(\lambda_{2}\) (which are the most IR-active ICMs of FEN and CAM) the carbonyl group is significantly involved: mode 1 has a large C=O stretching content, whereas mode 2 shows a significant C=O bending content. It is thus clear that the presence of carbonyl explains the stronger activity in the IR of FEN and CAM compared to MEFEN and MECAM (as expected). Turning now our attention to VCD, the ICM eigenvalues reported in Table 4 reveal, similar to IR, the presence in FEN and CAM of two significant positive eigenvalues (\(\lambda_{1},\lambda_{2}\)), which pair up with their negative counterparts \(\lambda_{6},\lambda_{5}\). The inspection of the associated eigenmodes (Supplementary Materials) reveals also for VCD the involvement of stretching/bending carbonyl motions occurring in-phase (\(\lambda_{1},\lambda_{2}\)) or out-of-phase (\(\lambda_{6},\lambda_{5}\)) with respect to skeletal distortions of the rest of the molecule. Therefore, based on the analysis of VCD-ICMs of the four molecules, we can provide a clue for the IR and VCD strengths of FEN and CAM being stronger than MEFEN and MECAM, as observed in [79]. Indeed, the direct involvement of carbonyl motions in the stronger ICMs is responsible for the observed differences in both IR and VCD.