Excitons in organic materials: revisiting old concepts with new insights

The development of advanced experimental and theoretical methods for the characterization of excitations in materials enables revisiting established concepts that are sometimes misleadingly transferred from one field to another without the necessary disclaimers. This is precisely the situation that occurs for excitons in organic materials: different states of matter and peculiarities related to their structural arrangements and their environment may substantially alter the nature of the photo-induced excited states compared to inorganic semiconductors for which the concept of an exciton was originally developed. Adopting the examples of tetracene and perfluorotetracene, in this review, we analyze the nature of the excitations in the isolated compounds in solution, in the crystalline materials, and in melt. Using single crystals or films with large crystalline domains enables polarization-resolved optical absorption measurements, and thus the determination of the energy and polarization of different excitons. These experiments are complemented by state-of-the-art first-principles calculations based on density-functional theory and many-body perturbation theory. The employed methodologies offer unprecedented insight into the optical response of the systems, allowing us to clarify the single-particle character of the excitations in isolated molecules and the collective nature of the electron–hole pairs in the aggregated phases. Our results reveal that the turning point between these two scenarios is the quantum-mechanical interactions between the molecules: when their wave-function distributions and the Coulomb interactions among them are explicitly described in the adopted theoretical scheme, the excitonic character of the optical transitions can be captured. Semi-classical models accounting only for electrostatic couplings between the photo-activated molecules and their environment are unable to reproduce these effects. The outcomes of this work offer a deeper understanding of excitations in organic semiconductors from both theoretical and experimental perspectives.


Introduction
For several decades, molecular materials have been under the spotlight as promising candidates for optoelectronic and photovoltaic applications thanks to their unique characteristics including lightweight, chemical tunability, and mechanical flexibility [1][2][3]. One of the reasons for their success is the possibility to tailor their electronic properties by synthetic means [4][5][6][7], thus avoiding the complex band engineering required for inorganic semiconductors [8,9]. However, the response of organic materials to external stimuli depends crucially on their state of matter. While gas phases and solutions typically bear signatures of single molecules, molecular aggregates reveal excitonic features [10]. Crystalline solids are particularly suitable for errors and not to the accuracy of the method [41] but such a lucky coincidence promotes a practice that can become misleading if not properly handled.
The time-dependent extension of DFT, TDDFT [42], has become an extremely popular method to calculate optical excitations in isolated molecules and molecular aggregates represented by non-periodic clusters. While preserving the computational efficiency of DFT, TDDFT provides access to the excited states. The coupling with effective solvation models has extended its range of applicability even to compounds in solution [43]. The ability of TDDFT to describe optical properties hits its limits when dealing with solids [41,44,45]. Issues stem from the incorrect long-range behavior of the Coulomb potential in the basic approximations for the exchange-correlation functional [46][47][48]. Range-separated hybrid functionals lead to notable improvements in the description of optical gaps and excitations [49,50], although they often require ad hoc tuning to grant good agreement with experiments.
The state-of-the-art ab initio methodology to describe excitons in solids is many-body perturbation theory (MBPT) [44], including the GW approximation for the electronic self-energy [51] to access quasi-particle band structures and the Bethe-Salpeter equation (BSE) [52] to calculate energies and wave functions of (bound) electron-hole pairs. This method was successfully applied to crystalline semiconductors and insulators for decades providing excellent agreement with experimental results [53]. GW/BSE calculations have recently established themselves also in the context of isolated molecules and clusters providing a good alternative to hybrid TDDFT [54][55][56][57][58][59][60]. Their added value is to offer, in addition to optical spectra, quantitative information about binding energies and spatial distribution of the excitations. The methodological and numerical complexity entailed by these calculations has so far prevented their development toward the inclusion, among other effects, of structural deformations, thermal effects, and coupling with vibrations. A few pioneering studies have explored these directions [61,62] even very recently [63,64], but they remain sporadic attempts. Similarly, schemes coupling GW/BSE with embedding methods are still in their infancy [65].
Despite the remarkable progress of ab initio theory to describe excitations in molecular materials, numerous ambiguities regarding organic materials have not been resolved yet. Previous experimental work has mostly been performed on fairly disordered molecular solids, while directionally and polarization-resolved optical measurements on highly ordered crystals are much rarer. In addition, there are only a few studies on the influence of polymorphism of molecular solids on their excitonic properties [66][67][68][69][70]. Currently, new molecular materials that promise high efficiencies in organic electronic devices are mainly studied, while moderate-sized molecular systems are considered to be fully explored and are therefore no longer examined. This is a somewhat unfortunate situation since the larger systems usually exhibit limited crystalline order and their size increases the computational costs for sophisticated computational approaches such as GW/BSE. On the other hand, detailed experimental data on supposedly well-known, highly ordered systems are required to validate existing concepts and explore new properties.
The established class of acenes is the ideal platform to address the above-mentioned issues. While these systems have been thoroughly studied [70][71][72][73][74][75][76][77][78][79][80][81], the influence of perfluorination on the structural and optoelectronic properties has been investigated mainly for perfluoropentacene (PFP) [82][83][84][85][86][87][88][89]. Compared to pentacene (PEN) and PFP, the smaller tetracene (TET) and its derivatives have a larger solubility and allow the growth of ordered crystals, which in turn make possible systematic direction-and polarization-resolved measurements. In addition, different polymorphs have been recently identified for perfluorotetracene (PFT) [90], which provide an interesting group of systems for a comprehensive analysis of their excitonic properties. Another important experimental aspect is the exciton dynamics due to singlet-exciton fission [91]. As this process is very fast for pentacenes, photoluminescence (PL) can be hardly observed in these systems, while the lower fission rate in the tetracenes allows also the observation of PL [92,93]. This has important consequences for a number of optical experiments. The somewhat smaller molecular size of TET, and thereby the lower number of electrons therein, enable a detailed optical analysis also of the often disregarded higher-energy excitations by means of the most accurate but also most computationally costly GW/BSE methods.
In this contribution, we present a comprehensive overview of the concept of excitons in organic materials taking tetracene and its perfluorinated variant as a platform for thorough investigation. After reviewing the adopted experimental and theoretical methods, we systematically analyze the optical response of these systems in different states of matter, ranging from isolated species in solution to crystalline phases and even melts, specifically addressing the requirements of sample quality and possible pitfalls in optical experiments. Through a detailed analysis of the electronic structure in the ground-and excited states performed with state-of-the-art first-principles methods, we provide insight into the optical absorption of these systems and the nature of the excitations therein. By studying their distribution and composition in terms of single-particle transitions, we are able to clarify when it is appropriate to speak of excitons and which characteristics allow their presence in the considered samples.

Systems
Among π-conjugated molecular materials, acenes have a number of electronic properties that make them prototypical. On the one hand, many of their properties, such as the energy of their molecular orbitals, and their thermodynamic characteristics such as, e.g. sublimation enthalpy or solubility, can be significantly tuned by the molecular size (i.e. the number of rings in their backbone); on the other hand, they all have very similar crystal structures with the characteristic herringbone packing motif. In addition, their molecular framework can be chemically substituted in many ways [94]. While flexible side groups have been used to improve the solubility of PEN such as in 6,13-bis(triisopropylsilylethynyl)pentacene, the packing motifs and intermolecular distances differ from those of the unsubstituted parental molecule, which limits direct comparison of the optical properties of corresponding solids [95]. In contrast, (per)fluorination of acenes notably affects the molecular energy levels and the effective charge distribution within the molecule (resulting, for instance, in the inversion of their quadrupole moments), but without significant changes in the molecular volume [96]. Moreover, it preserves the herringbone packing motif in the bulk crystal structure [97], thereby representing an ideal model system for detailed analyses. While the optoelectronic properties of PEN and PFP have been extensively investigated, comparative studies for their smaller analogs TET and PFT are very rare because the latter is not commercially available. Due to their higher solubility and diffusivity compared to pentacenes, tetracenes offer the advantage of simpler crystallization and thus enable the preparation of highly ordered crystalline samples.
In this work, we focus on TET and PFT, and investigate their optoelectronic properties in different states of matter: single molecules dissolved in solution, aggregated solids, and, for comparison, also a melt. Optical absorption measurements are performed using cuvettes (and different solvents) for solutions experiments and solid films with different degrees of crystalline order, as depicted in figures 1(a)-(d). The obtained results are often compared to those of other (perfluorinated) acenes to highlight systematic trends and/or specific properties. The molecular solids are realized with very different degrees of order, ranging from amorphous to textured (polycrystalline) films to single crystals (figures 1(c) and (d)). Depending on the degree of order of the films, directionally and polarization-resolved absorption measurements can also be carried out, which enable further analysis, e.g. of the polarization of the exciton. Although single crystals appear to be most suited to perform azimuthal and polarization-resolved absorption measurements (see figure 1(d)), the large photo-absorption cross-section of organic materials requires extremely thin crystals for transmission absorption measurements, which can be quite challenging. An alternative is given by films with large crystalline domains or ultrathin microcrystals, which can be analyzed locally by combining an optical polarizing microscope with a spectrometer to enable µ-spot spectroscopy (figure 1(e)). Such polarization-resolved absorption data provide experimental references that are analyzed in conjunction with first-principles results.
In the ab initio calculations, systems are modeled in three different states of matter: gas-phase molecules are assumed isolated and in vacuo (figure 1(f)); molecules in solution are simulated through the polarizable continuum model (PCM) [43] assuming a homogeneous cavity with the relative dielectric function (ϵ r ) of the solvent (figure 1(g)); finally, crystalline phases are computed assuming the molecules in the periodic arrangement given by the experimentally resolved lattice parameters ( figure 1(h)). For additional details, we redirect interested readers to a dedicated review [26]. In the following subsections, the methods adopted in the experimental and theoretical analysis are presented in detail.

Optical spectroscopy
The most commonly used method to characterize the electronic structure of molecules is optical absorption spectroscopy [98,99]. Here, not the energy of the single molecular states is measured, but excitations are detected, i.e. transitions between occupied and unoccupied molecular levels resulting from the absorption of specific light wavelengths. Since the low-energy electronic excitations of aromatic molecules are in the range of visible or UV light, this spectroscopy is also referred to as UV/vis absorption spectroscopy [98]. In contrast, level-selective spectroscopy such as UV photoelectron spectroscopy or inverse photoelectron spectroscopy is required to determine the absolute energy of occupied and unoccupied molecular levels [100]. However, it should be noted that the ionization potential and electron affinity determined in this way refer to ionized molecules while the optically excited molecules remain neutral. Compared to the characterization of films, the application of photoelectron spectroscopy to molecules in the gas phase is quite challenging, since the low density and the low photoemission yield result in very low intensities. Although molecules in the gas phase interact weakly with each other, which seems to be advantageous for an electronic characterization of single molecules, the high temperature of the vapor causes significant Doppler broadening and a considerable thermal population of vibrations that prevent a simple analysis. These problems are avoided with matrix isolation spectroscopy, in which the molecules to be examined are frozen out at cryogenic temperatures together with an inert noble gas matrix [101][102][103]. As a result, the molecules are very diluted so that they do not interact with each other and also do not exhibit a preferred orientation, but provide a sufficient density to enable precise measurements of absorption spectra, in which individual vibrational modes can also be resolved. The dielectric function of the matrix material (e.g. Ar, Ne, N 2 ) must also be taken into account here as it causes an energetic shift in the absorption bands, similar to liquids. Ultimate energy resolution can be achieved when molecules are embedded in supra-fluid helium nanodroplets using a seeded He-cluster beam, which allows for a Doppler-free spectroscopy and even enables the resolution of molecular rotational levels [104,105]. Another method commonly used in the chemical community to determine the energy levels of molecules is cyclic voltammetry. Although, strictly speaking, oxidation and reduction potentials are determined here, the energies of the frontier orbitals can be determined from this method with suitable referencing, but the values critically depend on reliable correction factors [106,107].
In view of the aforementioned difficulties in studying molecules in the gas phase, the optical properties of dissolved molecules are commonly examined instead. For this purpose, the molecules of interest are dissolved in a largely non-absorbing (at least within the spectral region of interest) solvent to measure the solution spectra. In this study, we used quartz glass cuvettes in combination with the UV transparent solvent dichloromethane (DCM), which enables measurements of absorption spectra in an energy range up to 6 eV. To analyze the optical properties of molecular solids, crystalline films grown by molecular beam deposition under high-vacuum conditions onto optically transparent substrates (quartz glass, KCl, or hBN) as well as ultrathin crystals (obtained from drop casting or liquid-assisted growth) were used. In combination with polarized light, this setup allows for polarization-resolved absorption measurements either by varying the angle of incidence or the azimuthal direction (see figures 1(c) and (d)) and, thus, to correlate the plane of polarization of the observed absorption bands with the molecular axes in oriented films or within a single crystal or domain.
The spectral absorbance A(E) is determined from the optical spectra recorded for (broadband) light passing the cuvette with the solution I sample (E), a reference spectrum of a cuvette filled with bare solvent (or the bare substrate without the film) I ref (E) and a background spectrum I dark (E) without illumination according to After illumination with UV light (here, E = 3.25 eV, λ = 380 nm), the spectral emission E is obtained from the sample spectrum I sample (E) after subtraction of the dark spectrum (without illumination) I dark (E): It should be noted that while spectrometers typically provide optical spectra as a function of wavelength (since the grating used fans out the light onto a detector array, whose angular positions scale with the wavelength), here, we present all the optical spectra in an energy scale for a direct comparison with the theoretical results. This step requires a corresponding Jacobian transform of the intensities since wavelength scales inversely with energy [108].

Experimental methods
While TET is commercially available (benz[b]anthracene Sigma Aldrich, purity 99.99%), PFT was synthesized and described in previous work [90]. For purification, the PFT was additionally re-sublimated before use. The organic thin films were grown under high vacuum conditions by means of organic molecular beam deposition (OMBD) from aluminum crucibles of resistively heated Knudsen cells. The film growth rates were monitored by a quartz crystal microbalance and typical rates of about 4 Å min −1 were used. Different optical transparent substrates were used including polished quartz glass slides (MicroChemicals) and graphene-coated quartz glass substrates (Graphenea, Spain). Additionally, KCl (100) substrates are used, which were prepared by cleaving slices of about 2 mm thickness from a single-crystal rod (Korth Kristalle GmbH) in air as well as hexagonal boron nitride (hBN) single crystals, which were prepared by mechanical exfoliation using an adhesive tape (SWT20, Nitto Denko Corp., for more details on their structural characterization see [109]). Before film deposition, all substrates were heated to 500 K in vacuo to remove residual adsorbates.
Large, millimeter-sized TET single crystals required for the structural characterization were grown by slowly cooling down a hot saturated solution in toluene over the course of two days. Ultrathin TET crystals suitable for transmission absorption spectroscopy were then prepared in two steps. After the growth of large crystals as described above, these crystals were partially re-dissolved by a few droplets of toluene, resulting in the recrystallization of ultrathin samples with thicknesses down to approximately 30 nm after evaporation of the solvent. We call this method recrystallization drop casting, as it is similar to the standard method of regular drop casting. Single crystals of PFT were obtained by means of liquid-assisted OMBD. In that process, which is described in more detail in the literature [110], a supersaturated solution of PFT is formed either in the ionic liquid 1-methyl-3-octylimidazoliumbis(trifluorometylsulfonyl)amide (Alfa Aesar) for the α-phase or silicone oil (VWR GmbH, 47 V 350 Rhodorsil) for the β-phase and kept at 40 • C, while continuously evaporating molecules into the solution [90]. After the deposition of several hundred nanometers at a small rate of 5 Å min −1 , isolated sufficiently thin crystals can be found and selected under the optical microscope.
X-ray diffraction (XRD) was utilized to characterize the crystalline structure and identify the phases of the films employing a Bruker D8 Discovery diffractometer using monochromatized Cu Kα radiation and a LynxEye silicon strip detector. Complementary, the film morphology was characterized by means of atomic force microscopy (AFM, Agilent SPM 5500), operated under ambient conditions in tapping mode using MikroMasch cantilevers with a spring constant of 40 N/m (f res = 325 kHz) and a tip radius of 8 nm. Additionally, the habit [111] of the single crystals was characterized by means of optical microscopy using a Nikon Eclipse LV-FMA polarization microscope (equipped with a sensitive DS-Ri2 camera with 14bit color depth per channel).
The optical UV/vis absorption spectra of the solutions and the molecular films were acquired in transmission geometry using an OceanOptics HDX-XR spectrometer in combination with a broadband halogen light source. Additionally, µ-spot UV/vis absorption spectra of single domains and small single crystals were recorded with an OceanOptics QE Pro spectrometer (spectral resolution ∆λ = 1.7 nm) that is attached to the above-mentioned optical microscope via a beam splitter. Using an additional polarization filter in the beam path of the microscope allows the recording of polarization-resolved optical absorption spectra in transillumination with a spatial resolution of better than 3 µm. Complementary, optical absorption spectra were also measured for heated films, that ultimately melt, which were sandwiched between two quartz glass slides and mounted between heatable copper plates while the temperature is monitored by a thermocouple (further details are given in the Supporting Information).

Theoretical methods
The first-principles calculations presented in this work were performed in the framework of DFT [112] and MBPT [51]. Assuming the electron density as the central quantity to access the properties of a many-body system, DFT is routinely implemented in the Kohn-Sham (KS) scheme [113] mapping the many-body problem into an auxiliary system of non-interacting particles obeying the Schrödinger equation in atomic units with the electron density evaluated from the KS wave-functions ϕ i (r) as The eigenvalues of equation (1), ϵ KS i , represent the energies of the single-particle KS states. For a deeper discussion about their meaning, we refer interested readers to dedicated work [114]. The effective potential contains the electron-nuclear Coulomb attraction (v ext ), the Hartree potential, v H [n](r), accounting for the mean-field repulsion of a single electron in the electronic density, and the exchange-correlation (xc) potential, v xc [n](r), including the remaining electron-electron interactions. The exact form of the last term is unknown and its approximations determine the accuracy of the obtained results. Local and semi-local approximations of v xc [n](r) are known to provide a poor description of fundamental gaps and electronic structures of highly inhomogeneous systems, such as isolated molecules, while their performance improves when dealing with extended molecular crystals. Hybrid functionals, including fractions of Hartree-Fock exchange, offer better accuracy. The subset of range-separated hybrid functionals-above all, CAM-B3LYP [115]-have gained increasing popularity for dealing with molecular systems. The GW approximation [51] is an established methodology to obtain accurate electronic structure including the quasi-particle correction. In this approach, the single-particle Green's function, where ϕ i (r) and ϵ KS i are the KS states and energies, respectively, obtained from equation (1), multiplies the dynamically screened Coulomb interaction to yield the electronic self-energy: Σ xc = iGW. This method is often applied in the so-called G 0 W 0 'single-shot' scheme: both G and W are evaluated once on top of KS results and included in the quasi-particle (QP) equation to correct the electronic energies. In equation (6), Z i is a renormalization factor introduced to compensate the evaluation of Σ xc at ϵ KS i by linear extrapolation. Due to the non-locality of the operators involved in the GW approximation, these calculations can become very expensive for systems composed of a large number of atoms such as molecular crystals. For this reason, a commonly adopted approximation is to skip the calculation of the electronic self-energy Σ xc and to mimic the QP correction to the electronic energies with a so-called scissors operator estimated, e.g., from experimental references of optical absorption onsets. This choice is well established in the community in case GW calculations are unfeasible for the systems under consideration [70,71,[116][117][118][119].
The BSE [52] is solved on top of the QP-corrected electronic structure to compute optical absorption spectra including (bound) electron-hole pairs. The problem is cast into an effective two-particle Schrödinger equation where o and u label occupied and unoccupied states, respectively. In the Tamm-Dancoff approximation, the BSE Hamiltonian for spin-unpolarized systems reads: The diagonal term,Ĥ diag , accounts for vertical transitions between occupied and unoccupied states,Ĥ dir is the direct electron-hole Coulomb attraction, andĤ x accounts for the repulsive electron-hole exchange, including the short-range part of the bare Coulomb potential,v C . The screened Coulomb interaction W in equation (9) is evaluated at ω = 0 (static screening) from equation (5). In equation (7), E λ are the excitation energies and A λ are the eigenvectors which contain information about the two-particle excitations. Both quantities enter the imaginary part of the macroscopic dielectric function which is used to describe the optical absorption spectrum of the material within the unit cell volume Ω. Equation (11) contains the transition-dipole moments expressed by in non-periodic systems, where r is the displacement operator, and by (13) in periodic systems, where the matrix elements are calculated for the momentum operator p. Since t λ is a vector, ϵ M (equation (11)) is a tensor with the number of non-vanishing, off-diagonal components determined by the crystal symmetries.
The BSE formalism provides quantitative information about the optical excitations of the systems. The binding energy of the excitons, E b , can be evaluated directly from the solution of equation (7) and the knowledge of the QP optical gap. In the context of inorganic semiconductors, where excitons are identified by narrow resonances below the absorption onset, the definition [120] is usually adopted. This definition is inappropriate for organic crystals, where the optical absorption spectrum is formed by distinct peaks even beyond the absorption onset, which characterize also the DOS [30,41,121]. In this context, a more appropriate definition of the exciton binding energy is where the independent (quasi)particle energies, E λ IPA , are the eigenvalues of equation (8) when the Coulomb interaction terms (equations (9) and (10)) are neglected. Solving equation (7) including only the diagonal term of the BSE Hamiltonian,Ĥ BSE ≡Ĥ diag , corresponds to the independent particle approximation (IPA). Adding only the screened electron-hole Coulomb attraction, H dir (see equation (9)) yields triplet excitations. While spin-forbidden and hence not directly detectable in optical absorption measurements from the ground state, triplet excitations may carry a finite transition-dipole moment and hence an associated oscillator strength.
Contrasting the corresponding spectrum against singlet excitations, obtained from the solution of equation (7) with the full BSE Hamiltonian (equation (8)), provides information about the influence of local-field effects (LFE), which are accounted for by the exchange term H x (equation (10)). LFE play a crucial role in spatially inhomogeneous systems such as organic materials, as extensively discussed in [41,48,116,121]. Finally, it should be noted that other techniques such as spin resonance and transient absorption measurements are able to probe triplet states making their theoretical identification particularly relevant also from an experimental perspective.
The solution of the BSE provides information about the character and the spatial distribution of the electron-hole pairs. The exciton wave function is a six-dimensional quantity where the BSE eigenvectors, A λ uok , weight each vertical transition from occupied to unoccupied states (note that we have introduced an explicit k-point dependence in the notation). Ψ λ (r h , r e ) is usually visualized by the isosurface of its square modulus, |Ψ λ (r h , r e )| 2 , representing the correlated electron (hole) probability density at a fixed position of the hole (electron). A complementary way to represent the spatial distribution of an exciton is given by the so-called exciton weights [122], i.e., the projections of the BSE eigenvectors, A λ uok , onto the band structure of the system. Hole and electron weights are defined as and respectively. The transition density is another useful quantity, established in quantum chemistry [123], that can be analyzed to gain insight into the character of the computed excitations. In a non-periodic system, it is expressed as where the BSE eigenvectors, A λ ou , multiply corresponding pairs of occupied and unoccupied states. Differently from the definitions of |Ψ λ (r h , r e )| 2 and w λ u/ok , in equation (17), A λ ou does not enter the expression with its square modulus. Hence, ρ λ TD (r) has positive and negative domains representing charge densities with opposite signs that provide information about the polarization of the excitation. In periodic systems, the transition density is less commonly used because it retains the periodicity of the single-particle wave functions and hence, in contrast to |Ψ λ (r h , r e )| 2 , it does not provide any information about the extension of the electron-hole pair.
An alternative route to compute optical excitations in the context of DFT is given by its TDDFT [42]. This approach is very popular in the treatment of non-periodic molecules and complexes and offers the advantage of an established interface with implicit solvation schemes, such as the PCM [43], to deal with molecules in solution. Optical properties in TDDFT can be accessed either through the direct time-propagation of the time-dependent KS orbitals [124] or in linear response [125]. While the real-time flavor of TDDFT is non-perturbative and hence most suitable to access optical nonlinearities [126][127][128] and easily interfaced with the Ehrenfest molecular dynamics scheme for coherent electron-nuclear coupling [129,130], in this study, we adopt the linear-response approach in conjunction with the PCM to calculate the optical excitations of molecules in solution.

Computational details
The procedure adopted for the ab initio calculations presented in this work consists of the following steps: (i) structural optimization is performed with DFT to determine the equilibrium geometry in which all interatomic forces are minimized up to a certain threshold; (ii) self-consistent DFT calculations of the systems in their optimized geometries are performed to access their total energy as well as their single-particle electronic structure; (iii) excited-state calculations, carried out either from GW/BSE (isolated molecules and periodic crystals) or TDDFT+PCM (molecules in solution), offer insights into the spectra of excitations and their character.
For structural optimizations in vacuo (isolated molecules and crystalline phases), the FHI-aims code [131] was used, thanks to its efficient routines optimized for this task. For these calculations, the generalized-gradient approximation is adopted for the exchange-correlation functional, as implemented by Perdew, Burke, and Ernzerhof (PBE) [132]. The pairwise Tkatchenko-Scheffler scheme [133] is employed to take into account van der Waals interactions. Tight integration grids and TIER2 basis sets are adopted. The atomic positions are relaxed until the Hellmann-Feynman forces are smaller than 10 −3 eV Å −1 . For the crystal structures, a 6×6×4 k-point grid is used for sampling the Brillouin-zone (BZ).
DFT electronic structures and optical excitations are computed on top of the optimized structures using different codes specifically implemented for compounds in different states of matter. By adopting this strategy, we can take advantage of the most appropriate and efficient tools for the target systems. Excited-state calculations of isolated molecules in vacuo are performed at all steps with the code MOLGW [134] using only its internal routines and no other external script. MOLGW is specifically designed to investigate the spectroscopic properties of non-periodic systems. It implements Gaussian-type basis set [135] (cc-pVTZ is adopted here) in conjunction with the frozen-core approximation and the resolution-of-identity approximation [136]. Taking advantage of the rich library of exchange-correlations functionals integrated into this package, the range-separated hybrid functional CAM-B3LYP is adopted in the DFT step as a starting point for the subsequent G 0 W 0 and BSE calculations. This choice was proven particularly suited to quantitatively reproduce the optical absorption spectra of organic molecules [59,137,138]. In practice, the solutions of the KS equations (equation (1)) are used to construct the single-particle Green's function (equation (4)) and to evaluate the dielectric function entering the screened Coulomb potential W (equation (5)), in order to compute the self-energy Σ and to solve the QP equation (equation (6)). The QP-corrected KS states enter also the terms of the BSE Hamiltonian (equations (8)- (10)).
Molecules in solution are computed with the Gaussian16 package [139] which offers a very efficient implementation of (TD)DFT coupled with the PCM. Note that in spite of recent promising advances [65], an equally viable interface between PCM and GW/BSE is currently not available and hence, we have to rely on TDDFT including the range-separated hybrid functional CAM-B3LYP to study optical excitations in solvated molecules. The static dielectric constants, ϵ r , are adopted to account for the considered solvents implicitly: ϵ r = 2.38 for toluene, ϵ r = 9.1 for dichloromethane, and ϵ r = 37.5 for acetonitrile are the default values in Gaussian 16 [139] employed herein. Input structures are optimized with the PBE functional [132] until interatomic forces are smaller than 0.005 eV Å −1 . We checked that this approximation for v xc does not affect the geometries compared to CAM-B3LYP adopted for the calculations of the optical properties for consistency with the GW/BSE simulations on the molecules in vacuo. For the same reason, the cc-pVTZ basis set was employed also in the TDDFT+PCM calculations of solvated molecules.
To describe the periodic molecular crystals, exciting [140], an all-electron full-potential code implementing the family of linearized augmented plane-wave plus local orbitals (LAPW+LO) methods, was adopted at all steps, including DFT and MBPT (i.e. the solution of the BSE), without the aid of any external routine. In the DFT runs, muffin-tin (MT) radii of 1.05 bohr, 0.7 bohr, and 1.4 bohr are chosen for carbon, hydrogen, and fluorine atoms, respectively; the cutoff for the plane-wave part of the basis is set to R MT G max = 3.3 for TET and naphthalene (NAP), and R MT G max = 5.0 for PFT, where R MT refers to the smallest MT radius of the entire system. The BZ is sampled by an 8 × 6 × 5 k-mesh for NAP, by a 6 × 5 × 4 k-mesh for TET, and 8 × 6 × 3 k-mesh for the two considered PFT polymorphs. The local density approximation in the Perdew-Wang implementation [141] is used to approximate the exchange-correlation potential. In the excited-state calculations for the crystals, the QP correction is mimicked by a scissors operator estimated by aligning the lowest-energy excitation in the computed spectra with the corresponding results from the measurements reported in this work. No empirical data or input from experiments was used in the actual calculations, ensuring that their ab initio nature is preserved. The scissors operator enters the denominator of equation (13) and the eigenvalues of equation (7) merely as an additive term and, as such, does not affect the ab initio character of the results obtained by solving the BSE as constructed from equations (8)-(10) [142]. Note that also in the calculation of x-ray absorption spectra solving the BSE, the G 0 W 0 step is usually skipped [143][144][145][146][147], due to the poor performance of this approach to correct core electron energies. In the construction of the BSE, which is performed using the DFT results obtained from the same code in a previous run, the screened Coulomb potential W is computed including 200 empty bands and an energy threshold for the local field effects equal to 1.0 Ha. The aforementioned k-meshes, shifted from the Γ-point, are taken for each system along with 20 occupied and 20 unoccupied bands.
For the specific characteristics and the technical usage of the codes adopted in this work, we refer interested readers to the corresponding user manuals. Input and output files of the calculations presented herein are publicly available free of charge (see Data Availability Statement below).

Isolated molecules
We start our analysis with the absorption spectra of the isolated TET and PFT molecules. Experimentally, they are taken for the moieties dissolved in dichloromethane (CH 2 Cl 2 ) while in the calculations, we initially consider the two molecules in vacuo (ϵ r = 1). The measured absorbance (figures 2(a) and (b), blue curves) reveals a number of distinct absorption peaks. The lowest-energy excitations of TET and PFT are found at 2.59 and 2.40 eV, respectively. In the spectrum of TET, the vibrational replicas of the first absorption peak are clearly visible in figure 2(a). Such a progression is often attributed to discrete molecular vibrational modes but it actually reflects local maxima of the vibronic density of states (DOS), where all vibrational modes and their oscillator strengths have to be considered [148,149]. The PL measurements (figures 2(a) and (b), red curves) reveal a small shift of the highest energy emission signal (∆E = 20 meV for TET and ∆E = 80 meV for PFT) with respect to the lowest energy absorption bands. This phenomenon, which is referred to as Stokes shift, is caused by the slight structural relaxation of the molecule in the excited state. This almost spectral overlap of the absorption and emission bands shows that the lowest-energy band actually corresponds to the (0,0) transition and thus to the optical band gap.
To gain insight into the origin of the absorption peaks in the spectra of TET and PFT, we analyze the spectra computed from first principles within the GW/BSE formalism (figures 2(c) and (d)). First and foremost, we appreciate the good agreement with the absorption measurements. The accordance between calculations and experiments is particularly evident in the UV region, where the energy of the strong resonance around 4.5 eV is almost identically reproduced in both datasets. The energy of the first maximum in the visible region is underestimated by 200 meV in the calculations performed on both TET and PFT. This value is on the order of typical deviations due to the flavor of the GW calculation, the adopted basis set, the underlying exchange-correlation functional, and/or the application of the Tamm-Dancoff approximation in the BSE calculations [150][151][152]. Additionally, it should be noted that ab initio results do not feature vibronic replicas as expected, since electron-vibrational couplings are not included in the calculations. Energy shifts with respect to the experimental data are attributed to the absence of polarizable media in these calculations, as discussed in detail below.
The first excitation in both spectra, labeled P 1 , comes from the transition between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) visualized in figure 2(e). The sizeable discrepancy between the energy difference of the frontier orbitals (5.79 eV in TET and 5.55 eV in PFT) and the energy of P 1 (2.41 eV in TET and 2.19 eV in PFT, see tables S3 and S4) results from the combined effect of the QP correction, introduced with the GW approximation and enhancing the fundamental gap, and the electron-hole attraction captured by the BSE, which shifts the absorption peak to lower energies [41]. In isolated molecules, especially with their ground state described by the CAM-B3LYP functional, the latter effect dominates over the former and leads to a considerable reduction of the optical gap. The excitation P 1 is polarized along the short molecular axis M (see reference framework in the inset), as shown in figure 2(f) by the plot of the transition density computed according to equation (17): notice the alternation of positive and negative charge-density domains only along the M-axis. The M-polarization of the lowest-energy excitation is a common feature of all acene molecules and it is responsible for the relatively weak intensity of the lowest-energy absorption maximum in comparison, for example, with the oligothiophenes, where the HOMO→LUMO transition is polarized along the long molecular axis [41,153]. The second excitation, P 2 , receives contributions from two single-particle transitions: HOMO-1→LUMO and HOMO→LUMO+1. Notice that, again, the excitation energy is significantly lower than the energy difference between the contributing orbitals, due to the electron-hole attraction captured by the BSE. In PFT, our calculations feature the HOMO-2 with the same symmetry as the HOMO-1 in TET (see figure 2(e)) and, as such, this state participates in the aforementioned transition to the LUMO. The contributions by two transitions indicate the existence of correlation in the optical excitations even of isolated molecules. Although these effects are less pronounced than in the corresponding crystals, as discussed extensively in the next sections, our results clarify that assuming transition energies as energy differences between orbital levels would deliver an inaccurate picture. It is also worth stressing that, while in TET these two transitions have almost equal weight, in PFT, HOMO→LUMO+1 provides the dominant contribution (see Supporting Information, tables S3 and S4). This subtle difference explains the non-identical transition-density distribution of P 2 in the two molecules (see figure 2(f)) and its substantially larger oscillator strength in the spectrum of PFT compared to TET (figures 2(c) and (d)). The opposite charge domains, oriented along the long molecular axis L, cancel almost entirely in the hydrogenated compound, in contrast to the perfluorinated one. The increasing intensity of this excitation in the spectra of the perfluorinated species is typical of the acene series [90] (see figure S1) and, as such, it can be considered a distinct feature of these compounds.
The higher-energy excitation, P 3 , which is found around 4.5 eV in both spectra, is the most intense resonance in the energy window displayed in figures 2(a)-(d). Its large oscillator strength is due to its polarization along the long molecular axis L. Its composition and transition density resemble those of P 2 (figure 2(f) and tables S3 and S4): here, however, charge-density-domain interference is constructive. A similar situation is again general to the acenes and can be explained by Hückel theory [90,154] or even with a more simplistic model (see figure S3 and related discussion in the Supporting Information). The last maximum visible in figures 2(a)-(d) is labeled P 4 . It is a weak excitation polarized along the short molecular axis M, predominantly stemming from the transition between the HOMO-1 (HOMO-2 in PFT) and the LUMO+1. In the experimental spectrum of PFT, it is visible as a shoulder of P 3 while it appears as a full peak in the one of TET. In addition to the bright excitations analyzed so far, it is worth mentioning that a number of dark states are revealed by the GW/BSE result (see tables S3 and S4). These excitations are forbidden for symmetry reasons, giving rise to destructive interference of the orbital contributions.
The similarity between the spectra of TET and PFT is not surprising and is common to other members of the acene family [90,154] (see figure S1). When comparing the energies of the lowest M-and L-polarized excitations among the oligoacene molecules, a monotonic decrease with the increasing number of rings can be noticed also for longer species [155]. The trends shown in figure S1 are, however, not identical. The energies of the M-polarized excitations scale faster than those of the L-polarized ones at least in the considered range of oligomer lengths between two and six rings. The different orbital energies reported in figure 2(e) should not surprise the readers. Instead, their almost rigid downshift should be noticed and understood as the effect of an electron-withdrawing functionalization of the conjugated carbon network due to fluorination. A systematic analysis of these effects has been reported in the context of graphene nanoflakes [156,157] and other fluorinated carbon-conjugated molecules [154,158].
To  note that also the PL spectra are subject to this effect (see figure S4). In addition to the energetic position of the various absorption bands, their intensity ratio also provides an important parameter that can be compared with the calculated spectra. In this context, however, it must be considered that optical spectrometers have limited dynamics and, due to the strong absorbance of molecular materials, the absorption spectra quickly saturate, which leads to a limitation of the observable peak heights and also causes an apparent broadening of the absorption bands. Here, it is advisable to analyze concentrationdependent series in order to rule out possible artifacts, as demonstrated in the case of the readily soluble anthracene (see figure S5).
The ab initio results obtained for TET and PFT in different solvents are reported in figures 3(c) and (d). We recall that in the methodology adopted for these calculations (TDDFT+PCM), the solvent polarity is reflected in the dielectric constant of the solvent. Although polarity and dielectric constant are, strictly speaking, different quantities, the knowledge that in a solvent the higher the solvent polarity, the larger ϵ r , is sufficient for the purpose of the following discussion (for further details, see [159]). The energy range considered in figure 3 includes only the lowest-energy excitation of TET, labeled P 1 in figure 2, whose energy matches very well the experimental result. Further details are provided in tables S8-S10. The vibronic replicas that are measured in association with this electronic transition are absent in the calculation due to the lack of electron-vibrational couplings in the adopted formalism. In the spectrum of PFT, we identify the first and second excitations, P 1 and P 2 (further details in tables S11-S13), previously analyzed for the single molecule in vacuo (see figure 2 and table S4). Again, the agreement between the calculated and measured energy of P 1 is excellent, while in figure 3(d), the second peak is blue-shifted by about 350 meV with respect to the experimental counterpart at 3 eV (see figure 3(b), noting that the first and second electronic excitations are followed by their vibronic replicas). In comparison with the results of the molecules in vacuo obtained from GW/BSE (see figure 2), the relative energy difference between P 1 and P 2 is still approximately 1 eV (see figure 3(d). From TDDFT+PCM, the spectrum is overall blue-shifted by about 400 meV such that the agreement with the experimental energy of P 1 is enhanced at the expense of P 2 . These discrepancies are within the expected range of variations due to the adopted approximations and computational parameters for these calculations [150]. On the positive side, it should be emphasized that the relative oscillator strengths between P 1 and P 2 in the spectrum of PFT are consistent with the experimental result. Likewise, the solvatochromic shift is correctly reproduced by the calculations, confirming the established success of the PCM in capturing these effects. A dedicated analysis of the effects of solvations in organic semiconductors captured by TDDFT+PCM can be found elsewhere [160]. To conclude this section, we remark on the excellent agreement between the TDDFT spectra in solution reported in figures 3(c) and (d) and the one obtained in vacuo from GW/BSE (figures 2(c) and (d)) net of the solvatochromic shift. The reason behind this accordance is given by the same parameters adopted in the underlying DFT calculations (basis set and exchange-correlation functional) as confirmed by the comparison of the orbital energies reported in tables S1 and S2 for molecules in vacuo and in tables S6 and S7 for those in solution; the spatial distribution of the orbitals is in both cases the same as reported in figure 2(e). On top of this, TDDFT mimics the mutually opposing actions of the QP correction to the gap estimated from the GW approximation and the electron-hole attraction computed from the BSE [41]. TDDFT in conjunction with hybrid functionals has been demonstrated to provide accurate results [161], especially for large molecules [162] and compounds characterized by charge-transfer excitations [163].
Finally, a general aspect in the optical characterization of molecular materials should be pointed out, namely their purity, which appears particularly critical for newly synthesized materials whose purity is not known a priori. Characteristic signatures of impurities are washed out of low-energy absorption edges. In the case of contaminants with less extended conjugate regions, these typically show higher energetic PL signatures, which appear to be more sensitive to contaminations than the absorption spectra, as shown by comparison of UV/vis and PL spectra of PFT raw material and sublimation-purified PFT (see figure S6).

Crystal phases
Next, we turn to the optical excitations of the molecular solids. Figures 4(a) and (b) compares absorption spectra of solutions and 30 nm thick films of TET and PFT. Analogously to the quartz cuvettes of the solution measurements, polished quartz glass substrates were used here, since they enable the recording of absorption measurements in transmission geometry over a large spectral range (up to 6 eV). Both molecular films reveal the presence of a new low-energy absorption band around 2.3-2.4 eV due to excitations of the lowest energy excitons that do not occur in solutions, where the lowest-energy excitations are around 2.5-2.6 eV. Other characteristic features of the solid-state absorption spectra are strongly damped vibrational progressions and distinctly broader resonances at higher energies. In particular, for energies around 4.5 eV, where the solutions show the most intense excitation, only a very broad and weak excitation band can be seen, while a more intense high-energy band around 5.4 eV occurs. In the case of TET, the low-energy excitation shows a clear splitting into sublevels due to the Davydov components, which will be analyzed in more detail later, while for PFT films such a splitting cannot be resolved. Complementary XRD measurements (figure S14) reveal that TET and PFT form polycrystalline films, where molecules adopt an upright orientation, characteristic of films of elongated molecules grown on inert substrates [164]. In this context, we would like to point out the different crystallographic indexing of the TET and PFT films. While such TET (and similarly PEN) films are described as (001)-oriented films, the PFT (and similarly PFP) films are described as (100)-oriented films, which is due to slight differences in the symmetry of both crystal structures. TET has a triclinic lattice with P1 symmetry, with the c axis chosen as the longest according to the axis convention (a < b < c). In contrast, the α and β phases of PFT form a monoclinic lattice with P2 1 /c symmetry. Since these crystal structures have additional glide planes, the corresponding axis is chosen as the c-axis by convention; hence, the a-axis is the longest one. A closer look at the bulk crystal structures of TET and PFT shows that the molecules in such TET(001) and PFT(100) planes adopt tilt angles of 21 • and 8 • with respect to the sample normal (see inset in figures 4(c) and (d)).
Although the molecules in such films are uniformly oriented upright, the crystalline domains are laterally isotropically distributed. Since their lateral extent is typically in the order of a few µm or less, the in-plane polarization of the optical excitations is usually not resolved and the absorption is domain-averaged. In order to enable a qualitative characterization of the polarization of the excitations and to distinguish whether they are polarized along the molecular L-axis or perpendicular to it, absorption spectra can be measured under different illumination angles (for technical details and the consideration of angular-dependent Fresnel factors, see [31]). As depicted in figures 4(c) and (d), absorption spectra recorded at normal and grazing incidence clearly show that the lowest energy excitations have the largest intensity at normal incidence, while the high energetic resonances, especially the one around 5.4 eV, are most intense for grazing incidence. As the absorbance depends on the relative orientation of the field vector E and the transition dipole moment d of the excitation according to |E · d| 2 , this indicates that the high-energy excitations are largely L-polarized. The same polarization is also observed for the absorption band around 2.85 eV that appears only for PFT, suggesting that it can be related to the other L-polarized excitation, P 2 , in the isolated molecules ( figure 2). Likewise, the most intense absorption band in the solid state around 5.4 eV is L-polarized, while the energetically similar single-molecule excitation (P 4 ) is M-polarized, showing that they are very different excitations despite the energetic proximity. Like many molecular solids, also acene crystals have a non-primitive basis. Consequently, the exciton degeneracy is lifted leading to the so-called Davydov-splitting of the lowest energy exciton due to the coupling of molecules in symmetry inequivalent positions within the unit cell [165]. The polarization of the resulting Davydov components was initially described by Kasha et al [22]  In order to resolve this splitting from polarization-resolved absorption measurements, sufficiently large single crystalline domains or single crystals are required. Since organic materials exhibit an exceptionally large photo-absorption cross-section, transmission absorption measurements on crystals require, extremely thin crystals, as discussed below. Instead, optical characterization is frequently done on the basis of ellipsometry and analyzing the reflected light [74,166]. An alternative approach is to use thin (semi-transparent) molecular films and combine optical microscopy with spectroscopy to enable µ-spot spectroscopy. This approach has been adopted previously for the case of pentacene, where a spot size of about 2-3 µm allowed polarization-resolved absorption measurements in transmission on individual single-crystalline domains [167].
Since many molecular films have little crystalline order or are even amorphous, the question arises, as to which excitonic signatures they have. To address this question, PEN was chosen as it shows the largest exciton binding energy and Davydov-splitting among the available acenes, and thin films were grown on quartz glass at cryogenic temperature (<150 K), which yields amorphous films [168]. The corresponding (room temperature) UV/vis absorption data reveal a somewhat broader but distinct excitonic excitation around 1.85 eV without any Davydov-splitting, in agreement with the absence of crystalline order (see figure S7). At this point, it should be noted that corresponding attempts to demonstrate this effect also for TET films were not successful because recrystallization takes place during thawing, while the amorphous film structure is only stable at low temperatures [169]. This underlines an important aspect in the optical analysis of molecular films, namely possible morphological and also structural changes that are driven by temperature. It is not the absolute temperature that is relevant here, but its value relative to the sublimation temperature, which can be only slightly above room temperature, especially for small molecules, where dewetting phenomena or recrystallization can occur. When examining molecular films and crystals optically, the fact that the molecular packing can change with temperature must be taken into account, too. Regardless of structural phase transitions (which occur, e.g., at 150 K in TET [169]), it should be noted that the thermal expansion in molecular solids is typically quite large and very anisotropic, due to the anisotropy of the , depend on the polarization, the inset reveals their opposed trend. In (c) a photoluminescence micrograph of a TET thin film (d = 300 nm) is shown. Since the emitted light passes a polarizer, this reveals the optical anisotropy of TET emission. A comparison of the absorption spectra of crystalline TET, PFT α-phase, and PFT β-phase is shown in panels (d), (e), and (f), respectively, together with the molecular orientation in the crystal as indicated by habit analyses (for (e) and (f)) and in-plane XRD measurements (for (d)). molecules, with both positive and negative expansion coefficients manifesting themselves [170], which can also lead to stress, especially in thin films, and also influences the measured exciton energies [171].
Another often overlooked difficulty encountered in the optical characterization of molecular solids is their emission, which can complicate the determination of the correct absorbance, especially for the low-energy exciton band. Systematic absorption measurements of TET films of different thicknesses show a supposed thickness dependence of both Davydov components, with the proportion of the lowest-energy component decreasing with increasing film thickness. In fact, this effect is caused by the spectral overlap with PL, the strongest emission of which comes from the lowest-energy Davydov component [172,173] (see figure S8 and related discussion in the Supporting Information). This effect is frequently overlooked as it does not occur in the most studied member of the acene family, namely PEN. The reason for this lies in the very high singlet fission rate in pentacene, which is why this material (at least at room temperature) shows practically no PL. In contrast, TET shows a significantly lower fission rate, which is why PL can be observed. The detailed analysis of the film thickness-dependent absorbance and PL allows the determination of the extinction (see figures S9-S11) and shows that only for TET films with a thickness of less than about 30 nm correct intensity ratio of both Davydov components is measured.
As mentioned before, in-plane polarization-resolved absorption measurements require thin films with large (rotational) domains or ultrathin crystals. Although thin molecular films can be easily deposited on transparent substrates (e.g. quartz glass), these usually only show very small domains and often a tendency of dewetting, such that the films break up into individual islands whose height is many times the nominal film thickness. A promising approach to overcome this limitation is the use of hetero-epitaxial growth of molecular films on inorganic transparent crystals, such as alkali halides (see, e.g. [86,174]), for which highly ordered surfaces can be easily prepared by cleavage. As shown in figure 5(a), this allows the preparation of thin and very homogeneous TET films on KCl(100) whose island thickness practically corresponds to the nominal film thickness. The PL micrograph reveals extended domains of more than 20 µm, which allow azimuthal polarization-resolved absorption measurements. However, a problem remains regarding the exact correlation between the azimuthal angles and the crystal axes in the crystalline domains.
Such an identification of the molecular crystal axes can be achieved by using molecular single crystals. For sufficiently large molecular crystals, in-plane XRD studies are possible, thus enabling the identification of the azimuthal crystal directions, which can also be correlated with the characteristic shape (the so-called habit [111]) of the crystals, as shown in figure 5 for the case of TET and PFT. Since these large crystals do not allow transmission absorption measurements, as discussed before, while smaller and thinner crystals (e.g. prepared by drop casting) do not allow XRD, habit analyses, which can also be complemented by polarization-resolved reflectance or PL spectroscopy, can be used to also identify the crystal axes in thin semi-transparent crystals suitable for a polarization-resolved µ-spot absorption measurements. However, due to the limited transmission of the imaging optics of the microscope, the accessible spectral range for such absorption measurements in the short-wave range is limited to 3.2 eV (380 nm). Figure 5 compares the polarization-resolved absorption spectra of thin TET and PFT (α-and β-phase) single crystals for polarizations where the two Davydov components are minimum and maximum in each case. As already found for the polycrystalline films, TET shows a clear Davydov splitting of 75 meV, while for PFT it is much smaller, about 45 meV) and seems almost identical for both phases despite the different number of molecules in the unit cell (β-phase: Z = 4, α-phase: Z = 2). The correlation with the crystal habit shows that the lowest energy Davydov component is polarized along the a-axis in TET and along the c-axis for PFT. However, if one considers the different conventions of axis designation, the same polarization results with regard to the molecular packing motif in TET and PFT.
The signatures of low-energy excitons in acene bulk polymorphs are very similar, as previously discussed in the context of PEN [70], since they all have very similar herringbone packing motifs and only the distances and angles change slightly [90]. In some cases, however, substrate-induced thin-film phases appear that show completely different packing motifs. A prominent example among acenes is PFP, for which a new interface-stabilized π-stacked polymorph manifests itself in films deposited on graphene or hBN, in which the molecules do not adopt a herringbone packing but are planar slip stacked [109,175]. Such a phase is also observed for the next smaller PFT in films grown on graphene or hBN (see [90] and complementary XRD data in figure S14). Figure 6 compares the optical absorption spectra of PFT films grown on bare quartz glass (blue curve), where the molecules are oriented upright and adopt the bulk phase, and on graphene-covered quartz glass (green curve), where the molecules lie flat and adopt the π-stacked phase (additional details on the structural analysis of PFT is provided in the Supporting Information, figure S14). Interestingly, the energy of the lowest energy exciton of π-stacked PFT is about 60 meV lower than for the bulk phase. A similar shift was observed before also for the next larger PFP [109] and reflects differences in the intermolecular coupling in both phases. In addition, one can see that the excitation around 2.8 eV is more intense for flat-lying molecules, thus reflecting the L-polarization of this band. On the other hand, the high-energy band around 5.5 eV is most intensely excited in films with standing molecular orientation, thus indicating the M-polarization of the underlying exciton. In order to get information about the in-plane polarization of the π-stacked films, freshly exfoliated hBN single crystals were used as a substrate, since they have a better quality than the graphene monolayers transferred to quartz glass. Although hBN is quite transparent, it exhibits a broad non-negligible absorption between 2.3 and 3.3 eV (see dashed lines in figure 6(b)). As shown by optical polarization microscopy (figure 6(c)), well-ordered PFT domains extending over more than 10 µm are formed and appear as rotational and mirror domains due to the three-fold symmetry of the hBN(0001) basal plane. Such domains are large enough to allow for the µ-spot in-plane polarization analysis which is shown in figure 6(b). Although the spectral range is limited to energies below 3.2 eV, the polarization-resolved absorption spectra on single rotational domains clearly confirm the previous polarization analysis. In particular, they show that the broader resonance found for PFT films on graphene around 3 eV is much sharper on hBN and has a clear in-plane contrast, with maximal absorbance when E is parallel to the long molecular axis, as for this condition the M-polarized low-energy is most weakly excited.
To complement the experimental analysis on the optical absorption of TET and PFT crystals, we analyze the ab initio results obtained from the solution of the BSE on top of DFT. We focus on TET and on two PFT phases, namely the α and the π-stacked polymorphs, having checked that the electronic structure of the β phase is very similar to the one of the α phase (see figure S15). Hence, we can expect that the two crystals will have an equally similar optical response. Notice that the presence of four inequivalent molecules in the unit cell of β-PFT would make corresponding BSE calculations extremely challenging. The high numerical costs that would prevent the necessary accuracy given the available computational resources are also the reason why the GW step is skipped for these calculations. The quasi-particle correction to the bands is mimicked by a scissors operator evaluated from the absorption onset of the corresponding experimental spectra. The adopted values are 1.86 eV for the TET crystal, 2.26 eV for PFT in the π-stacked configuration, and 2.33 eV for PFT in the α-phase (the complete list is reported in table S15).
In figure 7, from left to right, we show the spectrum of singlet and triplet excitations, as well as the one computed in the IPA. Notice that only the singlet spectrum can be compared with the experiment. Triplet excitations are spin-forbidden and hence not detectable upon absorption. However, one can compute their associated transition-dipole moments using equation (13) and hence obtain the corresponding oscillator strength. Triplet excitations are calculated by excluding the exchange term (equation (10)) in the construction of the BSE Hamiltonian (equation (8)). Hence, the comparison between singlet and triplet spectra computed with this method gives information about the role of the electron-hole exchange, which is a repulsive interaction due to the opposite sign of the two fermions. Physically, these couplings represent the effects of local fields which are known to be crucial in the spectra of inhomogeneous and anisotropic systems such as organic crystals [41,48,116,121]. Likewise, the IPA represents a theoretical construct in which all electron-hole Coulomb interactions are neglected. The analysis of corresponding spectra in comparison with singlet and triplet excitations provides information about the magnitude of these correlations.
We start by considering the singlet spectra calculated for TET and PFT crystals and visualized by the diagonal components of the imaginary part of its dielectric function (equation (11)). As shown in figure 7, left panel, in all systems, the lowest-energy excitation corresponds to a sharp resonance polarized along the , and results obtained in the independent-particle approximation (electron-hole attraction and exchange neglected). A Lorentzian broadening of 10 meV is applied to all the spectra. On the right-hand side, the ball-and-stick representations of the modeled crystal structures are shown: C atoms are depicted in grey, H atoms in white, and F atoms in green. Note that in these model structures, the crystalline axes are different for TET and PFT. crystal direction mostly aligned with the M-axis of the isolated molecules (figure 2). To highlight this connection, we name these excitations S M . Due to the presence of two symmetry-inequivalent molecules in the unit cell of each considered material, this excitation is split into two Davydov components separated from each other by 170 meV in the spectrum of TET and by about 60 meV in the one of α-PFT. The weaker Davydov component is labeled S M ′ . Notice that in the spectrum of TET it is at higher energy with respect to the more intense one, while in the result obtained for α-PFT it is at lower energies, according to the orientation of the molecules in the unit cell. In both crystals, they form a herringbone arrangement. However, in TET, the M-axis of the molecule has a larger component parallel to the lattice vector a which, in turn, is aligned to the x-axis (figure 7, right panel), compared to the component along b (almost parallel to the y-axis, see figure 5). It is worth recalling that, due to their non-cubic crystal structures, all systems considered in this analysis have non-zero off-diagonal components of the dielectric tensors, which are reported in full in figure S16. In α-PFT, the situation is different: The short axes of the molecules form an angle close to 45 • with respect to the (b, c)-plane. Hence, the first peaks in the yy-and zz-components of the imaginary part of the dielectric function have a similar magnitude with the latter being even lower in energy than the former. We ascribe this feature to the arrangement of the molecules in the unit cell. In the spectrum of π-stacked PFT, S M and S M ′ are energetically degenerate due to the absence of a herringbone angle between the molecules in this arrangement. Notice that the oscillator strength of S M , polarized along the y direction, is an order of magnitude larger than the intensity of S M ′ , polarized along z and thus representing only the residual contribution to the M-polarized excitation in the PFT molecules due to the slight tilt of the latter in the unit cell of the π-stacked phase.
Inspecting now the higher-energy region of the spectra, we find the counterparts of the L-polarized excitations in the isolated molecules (see figure 2). The strong resonance close to 5 eV in the spectrum of TET, labeled S L , is mostly polarized along the z direction along which the long molecular axis extends itself. Again, due to the complex arrangement of the molecules in the unit cell, even the yy component of Imϵ M contributes to the peak at 5 eV. The same feature can be identified also in the spectra of the two considered PFT polymorphs, whereby the polarization of S L is mainly along the x direction in both phases. In the π-stacked polymorph, the molecules are actually tilted in the (a, b)-plane of the crystal, thus activating also the yy-component of the dielectric function for S L . In the spectra of the examined PFT crystals, a weak, L-polarized excitation can be seen between S M and S L around 3.2 eV (see figure 7). This excitation, labeled S F L to highlight that it is optically active only in the spectra of fluorinated TET, is the direct counterpart of P 2 in the spectra of the isolated molecules ( figure 2). This transition, which is dark (bright) in the TET (PFT) molecules, remains such also in the crystal, suggesting once again the close correlation between the molecular origin of the optical transitions in isolated moieties and crystals. This being said substantial differences remain in the nature of the excitations in molecules and crystals.
Before diving into this crucial discussion, we complete the analysis of figure 7 by inspecting the triplet and IPA spectra. As clarified above, this comparison is exclusively meant to assess the role of the different Coulomb couplings acting between electrons and holes in the excitations of the considered TET and PFT crystals. To analyze the role of local-field effects in the optical response of these systems, we consider first the spectra of triplet excitations. The absence of the repulsive electron-hole exchange interaction is immediately detectable from the overall redshift of the excitations in comparison with the corresponding singlet spectra. The singlet and triplet absorption onsets differ by 0.67 eV in the results obtained for TET and by 0.87 eV in the spectra of both PFT phases, suggesting a larger electron-hole exchange repulsion in the latter crystals. This characteristic can be understood as a consequence of the perfluorination of the constituting molecules, which reduces the π-electron conjugation and hence introduces additional inhomogeneity in the system. It is worth underlining that in all systems, the polarization of the first triplet excitation is the same as the first singlet one, namely along the short molecular axis M; as such, it is named T M . The Davydov counterpart of T M , T M ′ , is present in all computed triplet spectra. However, its energy differs by a few tens of meV in the result obtained for TET and it is degenerate with T M in the spectra of both PFT polymorphs. This finding confirms that the splitting of these two components is triggered by the electron-hole Coulomb exchange. L-polarized excitations are present at higher energies also in the triplet spectra. As a common feature in all the systems considered in this analysis, a single (in the case of TET) or two sharp resonances (in both PFT crystals) appear in place of the broad absorption band that is visible in the UV region in the corresponding singlet spectra. Finally, in the triplet spectra of π-stacked and α-PFT, the counterpart of S F L is present, too, and labeled T F L to emphasize, again, that it is a specific feature of the fluorinated species. Notice that the oscillator strength of T F L relative to the neighboring peaks is lower than the one of S F L . This behavior is again typical of triplet spectra where the absence of electron-hole repulsion enhances the intensity of strong excitations and reduces the strength of the weak ones. Indeed, as extensively discussed in [41,48,121], local-field effects redistribute the spectral weight in addition to shifting it to higher energies.
Finally, we inspect the result obtained in the IPA ( figure 7). In this case, the electron-hole Coulomb interactions are completely neglected and the spectrum is given by the convolution of the matrix elements (equation (12)) computed for all vertical transitions assigned with equal weight. The difference between the IPA result and the spectra of singlet and triplet excitations is apparent. Not only is the first peak appearing up to 1 eV higher in energy in comparison to the singlets and around 1.5 eV with respect to the triplets (see  table S15 where these values, corresponding to the exciton binding energies, are reported); more strikingly, the onset is no longer characterized by sharp resonances but by a set of peaks forming almost an absorption band extending for approximately 1 eV. The polarization of the corresponding maxima is consistent with the results discussed for singlet and triplet excitations, although the excitonic nature of those peaks is, evidently, no longer captured. The strong peaks at higher energies are the counterparts of the L-polarized S L (T L ) resonances in the singlet (triplet) spectra. They remain sharp due to the localization of the transition within the single molecules in the unit cell. In principle, by taking the energy difference between these intense excitations around 5 eV in the IPA spectra and S L or T L , one obtains the corresponding binding energies. Remarkably, in all singlet spectra, S L is also very close if not even higher than 5 eV, suggesting a weakly bound character of this excitation (see figure 7). However, the inspection of the triplet spectra reveals that T L occurs around 3 eV in all systems, namely ∼2 eV below the IPA counterparts. From this analysis, we learn that the screened electron-hole Coulomb attraction red-shifts the L-polarized excitation in the crystal by about 2 eV but the local-field effects, due to the electron-hole repulsion almost entirely compensate for this effect. This intricate scenario shows once again how the interplay among the different flavors of quantum-mechanical Coulomb interactions in organic crystals affects their excitations.
The characteristics of the IPA spectra of the considered TET and PFT crystals shown in figure 7 may seem surprising, having in mind the typical shape of the joint DOS in inorganic crystals [53]. Why can a spectrum computed without excitonic effects exhibit sharp resonances even at high energies? To answer this question is it useful to inspect the plots of the density of states (DOS) computed for the three crystals. As shown in figure 8(a), the electronic structure of these systems is given by distinct peaks that are well separated from each other in energy [72,176]. It is evident that from such a distribution of electronic states, no continuum can be formed.
To deepen the analysis of the nature of the excitations in the optical spectra of the considered organic crystals, we inspect the distribution of the square moduli of the BSE eigenvectors (A λ ouk , see equation (7)) on top of the band structure of the systems (figure 9). To decipher these plots, we refer to the sketches depicted in figures 8(b) and (c) that schematically highlight the difference between a vertical transition between an occupied and an unoccupied state and an exciton. While the former is a single-particle entity representing de facto only a theoretical construct in a correlated material, the collective nature of the latter is apparent. An exciton receives contributions from several band-to-band transitions at different k-points and with different weights, held together by the electron-hole Coulomb interaction which is in turn mediated by the dielectric screening of the surrounding electron density. When these effects are negligible, such as in gas-phase atoms or molecules, the single-particle transition picture is valid. As discussed above, in TET and PFT, the first excitation is dominated by the HOMO→LUMO transition, thereby resembling in a molecular picture the scenario depicted in figure 8(b).
The results presented in figure 9 represent the distribution of the single-particle transitions contributing to the excitons highlighted in the spectra of TET and PFT crystals shown in figure 7. At a glance, we realize that all these graphs exhibit the pattern reported in figure 8(c). Starting from TET (figure 9, top row), we notice that the first exciton, S M , stems, as expected, from the transition between the valence-band maximum and the conduction-band minimum and their most closely surrounding states. Due to the presence of a non-primitive basis of two molecules in the unit cell of all considered crystals (see figure 7, right panel), the highest-occupied and the lowest-unoccupied bands are formed by two sub-bands corresponding to the HOMO and the LUMO of each molecule, respectively [176]. This sub-band splitting is responsible for the presence of two Davydov components in the optical spectra [165]. Indeed, the second excitation in the spectrum of TET, S M ′ , corresponding to the Davydov-split counterpart of S M , receives contributions from the same bands as S M , although with larger weights pertaining to the sub-bands more distant from the gap. The same result is obtained also for the two PFT polymorphs, with S M receiving contributions mainly from the valence-band top and the conduction-band bottom (see figure 9, middle and bottom left) and S M ′ stemming from transitions from the corresponding sub-bands. The composition of S F L in π-stacked and α-PFT is also closely related to its counterpart in the isolated molecule (P 2 ), with the contributions spread over the twofold bands stemming from the HOMO, the HOMO-2 as well as the LUMO and the LUMO+1.
The analysis of the higher-lying excitations reveals their excitonic nature, too, although with some remarkable differences with respect to the ones at the onset that enable us to elaborate on the concept of an exciton. In all the considered crystals, S L receives contributions from a number of bands in the valence and conduction regions (see figure 9, right panels). It is worth stressing that it is possible to capture this behavior only by solving the BSE where the screened electron-hole attraction (equation (9)) is computed for each pair of occupied-unoccupied states. The fact that different single-particle transitions contribute to the same exciton is a signature of the non-equal magnitude of this term for the various transitions. Recalling the composition of P 3 in the spectra of the isolated molecules (see figures 2(c) and (d) as well as tables S3 and S4), we should expect significant contributions from the second highest-and lowest-unoccupied band manifolds. Looking at figure 9, right panels, this appears only partially true, suggesting that correlation effects in the crystals promote the participation of a larger number of single-particle states in the excitations. This is a clear indication of the collective character of these excitations and hence, of their excitonic nature. Also, examining the spectra reported in figure 7, it is evident that a number of excitations build up the broader peaks in the UV region, including S L . In figure 9, right panels, we depict the lowest-energy member of this manifold, which we assume to be representative of it. However, we checked that the composition of the excitations in these manifolds is similar but not identical, further confirming the complex scenario disclosed by the study of the excited states in molecular crystals. The different k-space distribution exhibited by S L in TET, π-stacked PFT, and α-PFT suggest a different (de)localization of this excitation in real space. In TET, where the weights are mainly centered around the high-symmetry points V and R, we expect a larger spatial extension of the exciton. In both PFT polymorphs, the more homogeneous distribution of the weights over the entire BZ hints at a more localized character of S L in the unit cell of the crystals.
To better illustrate this concept, we have to turn to a smaller acene for which our computational resources enable a reliable description of the excitonic wave functions (equation (14)) in real space. Crystalline NAP, the two-ringed member of the acene family, is the ideal system to perform this analysis. Following the line of reasoning presented above for TET and PFT, we analyze the optical absorption of this system, the band contributions to selected excitations, and the corresponding wave-function distribution in real space (see figure 10).
In the spectrum of crystalline NAP (figure 10(a)), we immediately notice a difference with respect to TET and PFT: the lowest-energy excitation, S A , is extremely weak. Its polarization in the y-direction is consistent with the M-polarization in the isolated molecule [71]. However, the small aspect ratio of this molecule in comparison with TET and the orientation of the moieties in the unit cell with respect to the Cartesian system of reference, shift this excitation below the main peak, labeled S B . S B is polarized along y, too, with a small contribution from x, stemming from the M-polarized excitation in the spectrum of the isolated molecule (see figure S2). From the band contributions to the excitons ( figure 10(b)), we can clarify the peculiarity of NAP: S B stems mainly from the highest-occupied and lowest-unoccupied two-fold bands, while S A receives sizeable contributions also from the neighboring valence and conduction bands. Similar to TET, also in the spectrum of NAP the strongest peak in the UV-visible region corresponds to the L-polarized excitation in the single molecule which in the Cartesian system of reference adopted for the crystal is mainly oriented along z with small contributions from x and y (S C , see figure 10(a)). This excitation comes predominantly from transitions between the highest-occupied two-fold band and the second lowest-unoccupied one ( figure 10(b)). Given the small size of NAP, we can extend our analysis even higher in energy and discuss the next intense resonance (S D ), which is again polarized along y. The band contributions to this excitation involve states at the top of the valence region (highest and second highest twofold occupied bands) and in conduction above the lowest unoccupied states (see figure 10(b)).
This scenario is not dissimilar from the one discussed above for TET and the PFT polymorphs. Here, we can add the missing piece by inspecting the exciton distribution in real space. In figure 10(c), we visualize the electron probability correlated with the hole fixed in the position,r h , marked by the green dot, |Ψ(r h , r e )| 2 , see equation (14). We notice at a glance that in all the considered excitons, the electron can be found with equal probability on the molecule hosting the hole and on the neighboring ones. The wave function of S A is localized on the two molecules hosted in the unit cell. The same is true for S B although, in this case, the electron is spread also on a replica. S C exhibits a larger degree of real-space delocalization, with the correlated electron probability being non-vanishing even on the next-nearest neighboring molecules with respect to the one where the hole is fixed. Note that this characteristic can be traced back to the k-space representation of the exciton in figure 10(b), where the contributions coming from the second highest occupied two-fold band and those of the lowest unoccupied ones are remarkably distributed only on half of the BZ. Likewise, the pronounced delocalization of S D in reciprocal space is reflected in its enhanced real-space localization, with the electron density mostly focused on the molecule hosting the hole and spilling out to the symmetry-equivalent one along the b axis in the next unit cell (see figure 10(c)).
The last finding may look surprising at first, especially having in mind the textbook picture of excitons as sharp resonances below the absorption onset. This representation is correct, though, it strictly applies only to inorganic semiconductors where the bands form a continuum already very close to the frontier region, and the large dielectric screening limits the formation of bound electron-hole pairs to a few states within the optical gap. In organic crystals, the situation is very different. The electron states form bands, which, however, retain their similarity to the molecular orbitals. This peculiar electronic structure is clearly reflected in the DOS (see figure 8(a)), where a continuum does not appear even well above the gap region.

Discussion
Equipped with the knowledge gained so far, we want to come back to the concept of excitons in molecular materials and discuss the importance of the molecular environment for their formation. The comparison of absorption spectra of crystalline and amorphous films (see figure S7) shows a low-energy excitonic excitation for both solids, only the Davydov splitting is absent in the amorphous films due to the lack of crystalline order. This finding demonstrates that crystalline order is not required for the existence of excitons in molecular solids. In a simplified picture, optical excitations in molecules are sometimes denoted as 'excitons' even if they actually correspond to a HOMO→LUMO transition [177], which can be captured neglecting correlation effects and only accounting for the classical electrostatic interactions with the environment to reproduce the correct solvatochromic shift as shown in figure 3. On the other hand, an exciton is formed as soon as intermonomeric interactions become strong enough to enable the delocalization of the electronic wave functions around the excited molecule [26,79]. The precise packing motif is less important and only affects excitation energies usually within a range of a few tens of meV.
Based on this reasoning, excitons should exist also in a melt. In order to test this hypothesis, we measured the optical absorption spectra of a TET melt. A central problem in realizing these measurements is the requirement to have a very thin melt since µm-sized droplets are not transparent. For this purpose, we prepared an ultra-thin cavity consisting of two quartz glass slides housing a 40 nm thick TET film which is sealed through a previously sputtered gold frame. This stack was pressed between two heatable copper plates, each with a central opening permitting the transmission of light, and brought into the beam path of a UV/vis spectrometer (for details, see figure S17). Figure 11 shows the sequence of optical transmission absorption spectra recorded upon heating of the cavity from room temperature up to 360 • C. With increasing temperature, the initially (001)-oriented film becomes more disordered as evidenced by a disappearance of the Davydov splitting and an increasing absorbance of the L-polarized bands around 4.4 eV, while with increasing temperature the TET melts and forms a phase equilibrium with the gas phase. This explains the similar absorption signature of isolated molecules like for a dilute solution. When approaching the melting point of TET (357 • C), the lowest energy absorption band is red-shifted but still visible. This energy shift is attributed to the smaller density of the melt as compared to the solid film at room temperature, which increases the average intermolecular distance. After cooling down again to room temperature, the original absorption signature is essentially recovered. The small differences are attributed to the distinct dewetting occurring upon this heat treatment and the subsequent recrystallization, as evinced by complementary AFM and XRD measurements (see figure S17). Moreover, the comparison with the absorption spectrum of tetracenequinone (5,12-Naphthacenequinone, 97% purity, Sigma Aldrich) shows that oxidation of the TET can be safely ruled out during the heating series (see figure S18). These measurements show that low-energy excitonic excitations also occur in a melt which, in contrast to the solution experiments initially investigated, has identical solvent molecules to the solute.
This result emphasizes the importance of an environment of interacting (identical) molecules to generate an exciton. Only this characteristic enables the delocalization of the wave functions of the valence and conduction states contributing to the excitation and subsequently their condensation in a bound electron-hole pair. This is nicely visualized in NAP (figure 10(c)), where, regardless of the exciton energy, the electron density correlated to the (fixed) hole position is always (de)localized on the neighboring molecules, showing that the involved physical mechanisms are intrinsically quantum-mechanical and not merely electrostatic as a simple model based on dipole interaction could reproduce. This finding demonstrates that the idealized Frenkel picture in molecular materials should not be limited to one molecule but it requires the interactions between the specific moiety hosting the electron-hole pair and the other ones surrounding it. This intermonomeric coupling is quantum-mechanical in nature: it entails wave-function overlap and long-range Coulomb interactions that are dominant in a system characterized by an extended electronic distribution. Evidently, these effects are absent or marginal in an isolated molecule, i.e., in a molecule weakly coupled to an environment of other molecules, where the optically induced transition between occupied and unoccupied orbitals does not lead to the formation of an extended excitonic quasi-particle. Hence, a cluster of molecules, even if treated non-periodically, can be appropriate to describe excitons in organic materials as Figure 11. In-situ UV/Vis absorption spectra of a TET thin film (40 nm) while melting it by heating up to 360 • C inside a micro-cavity. long as the moieties in it are treated quantum mechanically. This is particularly the case when local interactions between the (usually heterogeneous) constituents rule the electronic and optical properties of the systems, such as in doped organic semiconductors [39,137,[178][179][180][181], where often samples do not present a long-range periodic arrangement.

Summary and conclusions
To summarize, we have provided a comprehensive experimental and first-principles analysis of the nature of the excitations in molecular materials, considering tetracene and its perfluorinated counterpart in different states of matter. In the isolated molecules, where correlation effects are mild, the single-particle character of the excitations prevails, and the lowest-energy excited states are dominated by one or very few occupiedunoccupied orbital transitions. No substantial differences are seen in relation to the solution environment which acts on the molecules mainly through electrostatic interactions. On the other hand, the excitonic character of the optical excitations in the solid phases is apparent. This is revealed experimentally through detailed polarization-resolved absorption measurements on single-crystalline samples, as well as theoretically from the solution of the BSE explicitly accounting for the periodicity of the wave functions. The latter formalism enables unprecedented insight into the nature of the excitations and the role of the different flavors of Coulomb correlations that act on the photo-excited systems. The electron-hole attraction builds up the excitonic resonance, picking up contributions from several band-to-band transitions over the entire Brillouin zone. The mutual repulsion between two fermions with opposite charges (i.e., the electron and the hole) distributes the oscillator strength among the optically active excitations and enables capturing the Davydov splitting that characterizes the spectrum of molecular crystals with two (or more) molecules at symmetry-inequivalent positions in the unit cell.
An important point highlighted by our analysis is that the bound character of the electron-hole pairs that are formed in a photoexcited organic crystal is not limited to the absorption onset or its closest vicinity: even excitations lying a few eV above the lowest-energy one exhibit similar localized character as the first excited states. This picture is opposite to the one known for inorganic semiconductors, where excitons correspond to sharp resonances within the optical gap following a Rydberg-like progression. The substantial differences between the excitons in inorganic and organic materials are intimately related to their respective electronic structure. While a continuum of bands characterizes the former in very close proximity to the frontier, in the latter, the electronic states remain discretely distributed while forming bands of varying dispersion according to the arrangement of the molecules in the unit cell. Experiments on molecular melts have revealed that excitons are formed even in these conditions when the ordered arrangement of the molecules has vanished but the moieties are still close enough to enable quantum-mechanical interactions such as wave-function overlap and Coulomb couplings among their electronic distributions. These findings are supported by calculations in which the effects of the environment are included only semi-classically, through electrostatic interactions with surrounding charges. In this situation, the results resemble those obtained for molecules in vacuo except for the correctly featured solvatochromic shift, suggesting that only the atomistic description of the moieties can capture the relevant quantum-mechanical effects mentioned above that grant the correct representation of excitons. Note that, due to the long-range nature of the Coulomb interaction in crystalline materials, these couplings may go well beyond distances between nearest-neighboring molecules. These effects are embedded in the periodic, Bloch-like description of the wave functions in organic crystals adopted herein. Non-periodic representations can still be able to capture the correct distribution of an exciton but the size of the cluster must be consistently converged.
To conclude, the faceted scenario depicted by our study suggests that the experimental and theoretical approaches adopted to analyze optical excitations in organic materials should be chosen appropriately in order to capture all the structural and electronic details that are crucial for the delivery of trustworthy results. The interest in organic crystals crossing the fields of physics, chemistry, and materials science leads to the use of different methods to address problems related to the photo-physics of these systems. While having adopted a solid-state physics perspective in this work, we do not intend to advocate for this approach and oppose other methods established, for example, in quantum-or computational chemistry. Our goal is to raise awareness that widespread concepts, such as excitons, may carry very different meanings in different contexts while being labeled with the same terminology.

Data availability statement
The data that support the findings of this study are openly available at the following DOI: 10.5281/zenodo. 7760270 (ab initio dataset) and 10.5281/zenodo.7779274 (experimental dataset).