On the other hand, other quantum methods such as Hartree-Fock do provide energy levels that can be interpreted as electron quasi-particle energies, i.e., the levels of an approximate freeparticle theory that effectively takes into account the many-body interactions between all electrons and nuclei. However, Hartree-Fock theory contains other limiting approximations and, unlike DFT, bond energies tend to be significantly underestimated. Hybrid functionals, which incorporate a portion of Hartree-Fock exact exchange in DFT calculations, produce a substantial improvement, for example in the predicted band gaps of semiconductors, though is still unreliable for metals and insulators (Paier et al., 2006; Jain et al., 2011).Therefore, electron transport results obtained this way using non-hybrid DFT functionals, such as LDA/GGA (local density approximation/ generalized gradient approximation), PBE (Perdew-Burke-Ernzerhof), or BLYP (Becke correlation functional; Lee, Yang, Parr electron exchange functional), despite being commonly used in the literature, are usually not expected to be very accurate. A double check using a hybrid functional, such as B3LYP, PBE0 and HSE (Heyd-Scuseria-Ernzerhof), though strictly required, unfortunately is not still frequently carried out.Another DFT limitation is the size of the system that can be simulated with present computers. Being an expensive

technique, typical simulations consider systems with a number of atoms in the order of magnitude of a hundred, which may be too small for the correct modeling of the material in disordered phases. As we discuss below, disordered phases play a key role in electron transport inside polymeric dielectrics.These accuracy and size limitations can be partially averted by using a one-electron theory in a mixed quantum-classical representation, in which a single electron is treated quantum mechanically, while the atoms interact classically, evolving in time according to the classical equations of motion. This method has been the most successful theoretical approach to explain spectroscopic and dynamics properties of hydrated electrons (Turi et al., 2012), and has already been applied to elucidate excess-electron transport properties in several representative motifs of polyethylene (Cubero et al., 2003a; Cubero et al., 2003b; Cubero et al., 2004; Wang et al., 2014). Polyethylene (PE), possessing a very simple chemical composition, with a single atomic group that is repeated into very long chains, [—(CH2)n-], and being widely employed in industrial applications, is the simplest example of a semicrystalline polymeric insulator.A key ingredient of the one-electron approach is the pseudopotential describing the interaction of the excess-electron and the atoms in the material, being obtained by either ab initio calculations (Turi et al., 2012) or fitting experimental data (Cubero et al., 2003a). The use of standard molecular dynamics techniques and the focus on just one excess electron allow the simulation of much larger systems than with DFT. It is also expected to be more accurate in PE due to the use of a semi-empirical pseudopotential. However, the development of a pseudopotential is not always easy to carry out for a generic electron-atom or hole-atom interaction, in contrast to DFT, which is not particular about the specific atoms forming the dielectric.In the following, we use PE as a case study to highlight the capabilities and limitations of current techniques. In Section 7.6, we study a more complex polymer insulator, showing that even though usual DFT methods may not be accurate, they can still serve as a useful tool to gain qualitative insight of electronic transport in polymeric dielectrics.