## 1 Introduction  
The steady and exciting development of quantum chemistry and, notably, density functional theory (DFT) methods has greatly enhanced our capabilities in understanding molecular spectra [1]. Current quantum chemistry computer codes can be used very reliably to assign the features observed in molecular spectra to electronic or vibrational transitions, with increasingly accurate levels of theory [2; 3; 4; 5; 6; 7; 8; 9]. Notable advances in tools available for computational spectroscopy have appeared in the recent literature [10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. These are commonly based on existing quantum chemistry engines, and their development often enhances the accuracy of the existing approaches. For instance, the harmonic approximation can be improved to deal with anharmonic effects [20; 21; 22; 23]. Within the rich and evolving scenario of computational spectroscopy, it is interesting to discuss approaches complementary to those most often employed to simulate spectra, which have been in use for a long time, and basically consist in weighted sums of selected lineshape functions [24] (ultimately, the weights of such sums are related to transition moments computed by quantum chemistry--see, for instance, [25]). In fact, one of the aims of the developments of our work is the embedding of the quantum-mechanical details of the chiroptical spectroscopic response of a molecule (which requires its complex electric dipole-magnetic dipole polarizability) within the simulation of the electromagnetic field distribution at the surface of plasmonic devices. Such simulations are instrumental to the interpretation of the experimental spectra measured from devices designed to enhance chiroptical interactions by the surface plasmon resonance of metal nanostructures [26]. The role of molecular polarizability is well recognized in the field of linear and non-linear spectroscopy, and led to the development of insightful computational tools for non-linear optics (e.g., [27; 28; 29]). Interestingly, polarizability was also introduced in the context of electronic CD aiming to calculate CD spectra of large molecules from the polarizability characterization of fragments [30]. We have investigated polarizability from the principles of quantum mechanics and light-matter interaction [31] adopting the notation used in [24]. The paper is organized as follows. We first illustrate the connection of CD with the polarizability tensor to obtain the relevant equations which relate the quantum mechanical transition rate to the trace of molecular polarizability (i.e., Equation (9)). Thereafter we show the use of the polar code, which has been written to implement such equations, for both electronic and vibrational transitions, based on the transition matrix elements obtained from quantum chemistry codes such as Gaussian [5]. For the benchmark of the electronic polarizability we consider a set of rigid molecules, thus avoiding conformational issues (which are irrelevant to the present study). Hence, \(\pi\)-conjugated molecules offer a rich playground, with reliable reference theoretical and experimental data available for comparison. For vibrational polarizability, we have considered another set of small and rigid chiral molecules, which have been very well characterized in the VCD literature, both experimentally and theoretically. In the final part of this work, we show how the Intensity Carrying Modes, which were pioneered by [PERSON] et al. in the context of vibrational polarizability and IR spectroscopy [32; 33], can be smoothly extended to VCD spectroscopy as well. Before describing our results, let us introduce the needed complex polarizabilities \(\mathbf{\alpha}^{em}\), \(\mathbf{\alpha}^{em}\), and \(\mathbf{\alpha}^{mm}\). This notation resembles the one adopted in theoretical investigations of antennas (e.g., [34]). It is also a notation that fits computer coding, and has fostered the development of the polar program.