## 3 Discussion
### Introducing Intensity-Carrying Modes in VCD
The APTs and AATs stored in the *.fchk file are used by polar to compute the dipole and rotatory strengths (Columns (b) and (c)) in atomic units (see Equation (14) and Section 4 for details), from which we computed the dissymmetry ratios \(g_{0k}=4R_{0k}/(cD_{0k})\) listed in Column (d)—please note the use of SI for the electromagnetic field, which implies the factor \(c=137.035999074\) in atomic units [41]. The dipole and rotatory strengths computed by Gaussian09 (Columns (e) and (f)) are used to compute the dissymmetry ratios \(g_{0k}=4R_{0k}/D_{0k}\) (cgs system) reported in Column (g). in the case of IR spectroscopy, can be expressed through the dipole derivatives with respect to the nuclear displacement Cartesian coordinates (\(x_{i}\); \(i=1...3N\)):  
\[\mathcal{M}_{ij}^{IR}=\sum_{k}\frac{\partial\mu_{k}}{\partial x_{i}}\frac{ \partial\mu_{k}}{\partial x_{j}}. \tag{16}\]  
For a given \(\mathbf{\mathcal{M}}\) matrix (either for IR, Raman, or ROA), the eigenvalue problem  
\[\mathbf{\mathcal{M}}\,\mathbf{c}_{k}=\lambda_{k}\mathbf{c}_{k} \tag{17}\]  
provides the eigenvalues \(\lambda_{k}\) associated with the maximized intensities, and the related nuclear displacements coefficients \(\mathbf{c}_{k}\) that describe a selected \(k\)th ICM [33; 86]. If one evaluates the dipole strength \(D_{0_{a}1_{a}}\) in terms of APTs (Equation (14)), one can easily justify the expression of the \(\mathbf{\mathcal{M}}\) matrix provided in Equation (16):  
\[D_{0_{a}1_{a}}=\langle 0_{a}|\mathbf{\mu}|1_{a}\rangle\cdot\langle 1_{a}|\mathbf{\mu}|0_{ a}\rangle=\frac{\hbar}{2\omega_{a}}\sum_{ij}L_{ia}\frac{\partial\mathbf{\mu}}{ \partial x_{i}}\cdot\frac{\partial\mathbf{\mu}}{\partial x_{j}}L_{ja}=\frac{ \hbar}{2\omega_{a}}\sum_{ij}L_{ia}\mathcal{M}_{ij}^{IR}L_{ja}, \tag{18}\]  
with  
\[\mathcal{M}_{ij}^{IR}=\frac{\partial\mathbf{\mu}}{\partial x_{i}}\cdot\frac{ \partial\mathbf{\mu}}{\partial x_{j}}.\]  
We mention that [PERSON]'s approach to ICM [33] makes use of internal coordinates instead of Cartesian nuclear displacements. This is just a matter of choice in the representation of the nuclear displacements, and it does not affect the conclusions reached in this Section. Equation (18) shows that the dipole strength of a given vibrational transition (\(0_{a}\to 1_{a}\)) is a quadratic form in the coefficients of the nuclear displacement of the associated normal mode (\(L_{ia}\)). Clearly, [PERSON]'s approach to ICM [33] seeks the eigenvalues of the \(\mathbf{\mathcal{M}}^{IR}\) matrix (Equation (16)), which is exactly the matrix of the quadratic form associated to the dipole strength reported in Equation (18). This observation justifies a similar approach for IR and VCD, provided that a suitable \(\mathbf{\mathcal{M}}\) matrix is defined also for the latter case. Following the strategy described for IR, to define ICM in VCD one may proceed by looking for a quadratic form associated with the rotatory strength \(R_{0_{a}1_{a}}\). By making use of Equation (14), one can express the rotatory strength in terms of APTs and AATs as follows:  
\[R_{0_{a}1_{a}}=\Im\left[\langle 0_{a}|\mathbf{\mu}|1_{a}\rangle \cdot\langle 1_{a}|\mathbf{\mu}|0_{a}\rangle\right]=\left[\left(\frac{\hbar}{2 \omega_{a}}\right)^{\frac{1}{2}}\sum_{i}\frac{\partial\mathbf{\mu}}{\partial x_{i }}L_{ia}\right]\cdot\left[\left(\frac{\hbar\omega_{a}}{2}\right)^{\frac{1}{2} }\sum_{j}\frac{\partial\mathbf{\mu}}{\partial x_{j}}L_{ja}\right]= \tag{19}\] \[=\frac{\hbar}{2}\sum_{ij}L_{ia}\left(\frac{\partial\mathbf{\mu}}{ \partial x_{i}}\cdot\frac{\partial\mathbf{\mu}}{\partial x_{j}}\right)L_{ja}.\]  
The last identity in Equation (19) provides the expression of the quadratic form suitable for VCD, characterized by a matrix \(\frac{\partial\mathbf{\mu}}{\partial x_{i}}\cdot\frac{\partial\mathbf{\mu}}{\partial x _{j}}\). We note that this matrix is not symmetric.