## 2 Results
For a molecule with eigenstates \(\left|k\right\rangle\) and transition energies \(\hbar\omega_{k}\) from the ground state \(\left|0\right\rangle\), the electric polarizability can be written as a sum over states, with opposite sign damping \(\Gamma\) in the resonance term (first term) and off-resonance term (second term) [35; 43; 44; 45]:  
\[\alpha_{ij}^{\alpha\alpha}(-\omega;\omega) =\frac{1}{\hbar}\sum_{k}\frac{\left\langle 0|\mu_{i}|k\right\rangle \left\langle k|\mu_{j}|0\right\rangle}{\omega_{k}-\omega-i\Gamma/2}+\frac{1}{ \hbar}\sum_{k}\frac{\left\langle 0|\mu_{j}|k\right\rangle\left\langle k|\mu_{i}|0 \right\rangle}{\omega_{k}+\omega+i\Gamma/2}= \tag{5}\] \[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]0By recognizing in the second line of Equation (5), the numerators as the outer products of the transition electric dipoles (namely, \(\mu_{0k}\otimes\mu_{k0}\) and \(\mu_{k0}\otimes\mu_{0k}\)), one can compactly write the polarizability tensor as follows:  
\[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]1  
By introducing the formal substitution (\(\mu_{i}\to\mu_{i}\), \(\mu_{j}\to m_{j}\)) in the expression of the electric polarizability given by Equation (5), one can devise the expression of the complex electric dipole-magnetic dipole polarizability \(\mathbf{\alpha}^{cm}\):  
\[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]2  
With a similar reasoning, the following expression can be written for the magnetic dipole polarizability tensor \(\mathbf{\alpha}^{mm}\):  
\[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]3  
Near resonance conditions with the \(0\to k\) transition, Equations (6)-(8) can be used to relate the trace of molecular polarizabilities to the transition rate given by Equation (3) (see Appendix A for details):  
\[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]4  
As well-known, we recall that the imaginary part of \(\mathbf{\alpha}^{\alpha e}\) relates to absorption phenomena, and the real part to dispersive ones. Because of the presence of the purely imaginary magnetic dipole transition moment in the expression of \(\mathbf{\alpha}^{cm}\), the absorption and dispersive terms in \(\mathbf{\alpha}^{cm}\) are the real and the imaginary parts of \(\mathbf{\alpha}^{cm}\), respectively (see also Equation (A7) in Appendix A). When neglecting vibronic effects, the \(0\to k\) transitions introduced in Equations (6)-(8) are taken as vertical electronic transitions, and they can be conveniently computed by, e.g., TDDFT. This is the approach adopted by the polar code. In summary, by the calculation of frequency-dependent polarizabilities, Equation (9) can be used:  
1. to simulate UV-Vis absorption spectra in the absence of vibronic effects: \[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]5 (10)
2. to simulate electronic CD (ECD) spectra [24]: \[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]6 (11) and
3. to simulate the dissymmetry factor \(g\), which is the ratio of CD (\(\Delta e\propto w^{CD}\)) to ordinary absorption (\(\epsilon\propto w^{abs}\)) [24]: \[\tilde{D}_{0k}=\mathbf{m}_{0k}\cdot\mathbf{m}_{k0}=\left(D_{0k},0\right)\quad\text{ electric dipole strength};\]7 (12)  
To simulate IR and VCD spectra, one has to evaluate the molecular polarizabilities by considering the \(0\to k\) transitions as vibrational. If one focuses on the vibrational transitions in the mid-IRrange, within the double harmonic approximation, the sum over states is restricted to one-quantum transitions \(0_{a}\to 1_{a}\) over the 3N-6 vibrational modes of the system, based on mechanical harmonicity.