Organic semiconductors include two major classes of materials: molecular crystals of small pi-conjugated molecules (to which we shall mainly refer in the following), and polymeric semiconductors. The molecular crystals are stabilized by long range intermolecular Van der Waals (vdW) dispersion interactions. Even though these vdW forces are weak, their influence on the electronic and optical properties has been shown to be relevant for several materials. In particular, since the charge transport mobility of organic semiconductors is closely related to the interactions between the molecules of the crystals, the crystal structure is a critical factor in determining the overall transport properties. Standard DFT approaches do not properly account for weak dispersive forces, which stem from the non-local part of the correlation energy. For this reason, experimental values of the structural parameters are often used in the calculations (rather than theoretically optimized values as it is generally the case for inorganic crystals) [14]. However, this situation may change in the near future, as recent developments offer the possibility of an accurate description of vdW interactions within DFT via the addition either of a non local correlation term in the exchange-correlation functional [15] or of empirical correction terms to describe the non-local correlation [16]. (Although in principle high-level quantum chemistry approaches, such as Møller-Plesset perturbation theory [17] and coupled-cluster approach [18,19], include such non-local effects, they are not computationally viable for the description of large aggregates or periodic systems.) It will be thus interesting to apply these novel approaches to organic crystals and see how a more accurate description of intermolecular interactions affects other computed physical properties, such as, e.g., the electronic structure and the electron-phonon couplings.

Numerous first principles DFT calculations of the electronic band structure of organic molecular crystals have been reported, including analyses of the dependences of the band dispersion on the crystal structure [20,21] and molecular chain length [22]. Whereas local and semi-local functionals are generally used in periodic DFT calculations, recently a few band structure calculations employing hybrid functionals [23] have been reported; this is in contrast to studies on isolated molecule models where hybrid B3LYP functionals have been extensively applied. Many-body GW studies of organic crystals are rare: they have been performed only in a few cases to obtain quasiparticles band structures and accurate values of the (HOMO-LUMO) energy gap [12,13].

The interfaces between the conducting electrodes and the organic semiconductors have a key role in the performance of organic devices [24]. In particular, transport across an interface crucially depends on the alignment of the energy levels at the two sides of the interface. Standard local and semi-local implementations of DFT have been widely applied to study organic- inorganic interfaces formed by either isolated organic molecules or organic monolayers on metal and semiconductor surfaces [25]. Even though trends, e.g. as a function of the metal workfunction, appear to be correctly captured by these calculations, the predicted energy level alignments are generally inaccurate in absolute value (see, e.g.[26]). To overcome these difficulties, different approaches have been explored, including accurate but cumbersome GW [11,12,27] and Quantum Monte Carlo [13] calculations, as well as computationally less demanding simplified GW calculations [28] and LSDA+U methods [29]; altogether, however,the number of such studies is still very limited.

Transport across metal-organic interfaces depends sensitively also on the distance between the metal and the semiconductor, i.e. on the adsorption configuration of the organic molecule on the metal substrate. Standard (semi-) local DFT describes well strong chemical bonds (chemisorption) but the bonding of pi-conjugated organic molecules with metal surfaces is often determined by weak dispersive forces. The challenge of describing such a weak adsorption (physisorption) situation is currently a topic of active research; proposed approaches include the use of (semi-)empirical models [30], the inclusion of nonlocal correlations into the DFT functional [31], and a combination of exact exchange and random-phase approximation [32].

The optical spectra of organic semiconductors show prominent excitonic effects. Unlike the situation in inorganic semiconductors, exciton binding energies in organic materials are quite large, ranging from a few tenths up to more than one eV [33]. Many-body perturbation theory (MBPT) calculations of optical properties and exciton binding energies have been performed for several organic crystals [34]. A possible alternative to the accurate but computationally cumbersome MBPT method is TDDFT, whose suitability to calculations of the optical properties of organic materials is currently under investigation [35]. Since the lifetime of excitons in organic materials is very long (a few orders of magnitude larger than that of inorganic materials) [36,37] while their dynamics is very slow, an important challenge is to describe the system under non-equilibrum for the long time period of the exciton lifetime .

The transport properties of organic semiconductors are characterized by carrier mobilities which are generally much lower (mu ~ 1 cm^2/ V s at room temperature) than in covalently bonded inorganic semiconductors. Such low mobilities are related to the narrow (~0.1 eV) electronic energy bands and the strong electron-phonon coupling in these materials [38]. It is widely accepted that the most adequate description of charge transport in crystalline organic semiconductors is based on the concept of polaron. Key parameters in this description are the electron-phonon and the electronic intermolecular couplings. [9,39]. The amount of theoretical work in this field is huge [9]. However, the detailed understanding of the polaron dynamics and its dependence on temperature remains as one of the most important challenges in the field of organic electronics [40].