## 2 Results  
The theoretical description of CD phenomena entails both electric and magnetic effects. In a detailed review [24], [PERSON] presented the theoretical description of circular dichroism and optical rotation for molecules in solution state. The central role is played by the perturbation operator \(V_{\pm}\) associated to left-handed (\(-\)) and right-handed (\(+\)) circularly polarized light, which is defined through the electric (\(\mathbf{\mu}\)) and the magnetic (\(\mathbf{m}\)) dipole operators. By making use of the results illustrated in Ref. [24], for a given transition between states \(0\) and \(k\), the expectation value of such operators is the following:  
\[\left|V_{\pm,0k}\right|^{2}=\frac{1}{3}\left(D_{0k}E_{0}^{2}+G_{0k}B_{0}^{2} \mp 2R_{0k}E_{0}B_{0}\right)=\frac{2}{3}\frac{I}{c\epsilon_{0}}\left(D_{0k}\mp \frac{2R_{0k}}{c}+\frac{G_{0k}}{c^{2}}\right); \tag{1}\]  
\[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]  
\[\tilde{G}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(G_{0k},0\right)\quad\text{ magnetic dipole strength};\]  
\[\tilde{R}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(0,R_{0k}\right)\quad\text{ rotatory strength},\]  
where \(E_{0}\) and \(B_{0}\) are the electric and magnetic field amplitudes of the impinging light wave, \(\mathbf{m}_{0k}=\left\langle 0|\mathbf{\mu}|k\right\rangle\) is the electric transition dipole moment, and \(\mathbf{m}_{0k}=\left\langle 0|\mathbf{m}|k\right\rangle\) is the magnetic transition dipole moment. \(I\) represents the light intensity, \(c\) is the speed of light, and \((a,b)\) indicates the complex number \(\tilde{z}=a+ib\). [PERSON]'s review [24] adopts cgs units. For the reasons discussed by [PERSON] [41], in this work we prefer using atomic units for the numerical values of polarizabilities and we consider the SI system for the description of the electromagnetic field. For the sake of simplicity, we disregard solvent effects and adopt electromagnetic waves _in vacuum_. Therefore, the intensity of the electromagnetic wave required by [PERSON]'s treatment (Equation (6) in Ref. [24]) is given by \(I=\frac{1}{2}c\epsilon_{0}\left|E_{0}\right|^{2}\), where the magnetic (\(B_{0}\)) and electric field (\(E_{0}\)) amplitudes are related one another by the relation \(B_{0}=E_{0}/c\)[42]. The electromagnetic energy absorbed per unit time as a consequence of the \(0\to k\) transitions promoted by the electromagnetic perturbation at frequency \(\omega\) is given by [24]:  
\[w_{0k}(\omega)=\frac{\pi}{2}\frac{\omega}{\hbar}|V_{0k}|^{2}\rho_{k}(\omega), \tag{2}\]  
where \(\rho_{k}(\omega)\) is the density of states for the \(0\to k\) transition, and it is normalized, i.e., \(\int d\omega\,\rho_{k}(\omega)=1\). By substituting Equation (1) into Equation (2), one obtains:  
\[w_{\pm,0k}(\omega)=\frac{\pi}{3}\frac{I}{c\epsilon_{0}}\frac{\omega}{\hbar} \rho_{k}(\omega)\left[D_{0k}\mp\frac{2R_{0k}}{c}+\frac{G_{0k}}{c^{2}}\right]. \tag{3}\]  
To connect Equation (3) with the molecular polarizability tensors, it is instrumental to write the inner products among the transition dipoles (Equation (1)) as traces of outer products:  
\[\tilde{D}_{0k}=Tr\left(\mathbf{\mu}_{0k}\otimes\mathbf{\mu}_{k0}\right), \tag{4}\] \[\tilde{G}_{0k}=Tr\left(\mathbf{m}_{0k}\otimes\mathbf{m}_{k0}\right),\] \[\tilde{R}_{0k}=Tr\left(\mathbf{\mu}_{0k}\otimes\mathbf{m}_{k0}\right).\]  
We recall that the tensor (outer) product \(\mathbf{a}\otimes\mathbf{b}\) of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) has matrix elements \(c_{ij}=a_{i}b_{j}\). Therefore, the dot product \(\mathbf{a}\cdot\mathbf{b}\) can be seen as \(\mathbf{a}\cdot\mathbf{b}=\sum_{i}a_{i}b_{i}=\sum_{i}c_{ii}=Tr(\mathbf{a}\otimes\mathbf{b})\).