Molecular vibrations reduce the maximum achievable photovoltage in organic solar cells.

The low-energy edge of optical absorption spectra is critical for the performance of solar cells, but is not well understood in the case of organic solar cells (OSCs). We study the microscopic origin of exciton bands in molecular blends and investigate their role in OSCs. We simulate the temperature dependence of the excitonic density of states and low-energy absorption features, including low-frequency molecular vibrations and multi-exciton hybridisation. For model donor-acceptor blends featuring charge-transfer excitons, our simulations agree very well with temperature-dependent experimental absorption spectra. We unveil that the quantum effect of zero-point vibrations, mediated by electron-phonon interaction, causes a substantial exciton bandwidth and reduces the open-circuit voltage, which is predicted from electronic and vibronic molecular parameters. This effect is surprisingly strong at room temperature and can substantially limit the OSC's efficiency. Strategies to reduce these vibration-induced voltage losses are discussed for a larger set of systems and different heterojunction geometries.


SM.1 Derivation of the theoretical approach
Hamiltonian -The electronic and vibrational properties of the studied systems as well as their interactions can be described by where The phonon part of the Hamiltonian is described by harmonic oscillators ph = ∑ ℏ � † + 1 2 � including low-frequency and high-frequency modes with their intramolecular electron-phonon coupling (EPC) el−ph according to Here, the orbital indexing follows the notation in Supplementary Equation 2 and denotes the vibration mode with mode energy ℏ and dimensionless coupling constants . The material parameters for the vibrational frequencies ℏ and the EPC constants of the intra-molecular modes are discussed in section SM.3 of the Supplementary Methods.
Henceforth we use the simplified notation † → † for indexes when refers to LUMO-type orbitals and † → ℎ † when refers to HOMO-type orbitals (for all molecular species), because the excitations of interest are the energetically lowest ones including Frenkel-type molecular excitons (ME) and chargetransfer (CT) excitons. We restrict our model to those excitations. More orbitals can be added between a LUMO-type electron and a HOMO-type hole is considered in a form that conserves the orbital , thus excluding effects of intrinsic exciton annihilation or creation. Furthermore, we consider the direct electron-electron interaction with respect to the inter- and the exchange-type terms ℎ ′ℎ′ = � ℎ ′ℎ′ ME ∀ℎ, with = ℎ and ∀ℎ ′ , ′ with ′ = ℎ ′ that are next neighbors of ℎ, 0 otherwise . (5) min is the distance between two next-neighbor molecules. CT is the electron-hole binding energy of two neighboring molecules that is calculated according to section SM.4 below. ME is the binding energy of the molecular exciton with a value discussed in section SM.2 below. Finally, ℎ ′ℎ′ ME denotes excitonic transfer integrals, which are typically dominated by the dipole-dipole interaction of transition dipoles at neighboring molecules. This contribution describes the delocalisation of Frenkel excitons and is assumed to be at most on the order of the electronic transfer integrals or below. Since these parameters depend on wave function overlap, analogous quantities for CT states are orders of magnitude smaller and can be neglected.
Exciton DOS -We turn to the description of the exciton DOS (EDOS) as defined in Equation 2 of the main text. For its calculation, we specify a subset of initial electron-phonon states for exciton absorption. For these initial states we consider product states | (0) ⟩� ph �, where the electron part is denoted by | (0) ⟩ and is the fully occupied set of HOMO-type orbitals (corresponding to the charge neutral configuration without thermal exciton occupation). For the phonon part, we use states | ph ⟩ that diagonalise in the neutral electronic ground state: As a result, the EDOS in Equation 2 reduces to To simplify the analysis regarding high-frequency vibrations, we split the set of modes into a group of lowfrequency vibrations ph lf that may be thermally occupied and a group of high-frequency vibrations ph hf whose Hilbert space is restricted to the respective vibrational ground state denoted as �0 ph hf �. As a result, the density operator ph of the initial vibrational state reads with ph lf the corresponding partition function and = ( B ) −1 . A more complex model for highfrequency modes is not necessary here.
Inserting ph into the EDOS yields with the electro-vibronic ground state |0⟩ ev = � (0) ��0 ph hf � and the remaining trace Tr ph lf [… ] = ∑ ⟨ ph lf, | … | ph lf, ⟩ over the states describing low-frequency vibrations � ph lf, � which can be written as We can approximate the time evolution operator , which holds if the corresponding commutator vanishes or if the correction terms are small. This is realised when the intrinsic time scales of − ph lf are much smaller than the time scale of ph lf . We obtain where el (0) + ph hf,(0) can be defined as suitable energy zero in the excitation process.
Tracing up over the slow modes yields the EDOS (see Equation 3 of the main manuscript) ( ) depends on the microscopic molecular parameters (EPC parameters ℎℎ and ) and the mode energies ℏ . The thermal disorder in Supplementary Equation 14 is induced by slow vibrations which contribute independently through a set of Gaussian random variables … for each mode . 〈 … 〉 is the Bose-Einstein distribution function. We emphasise that this analytical result is exact in the indicated limit of low vibrational frequencies.
Numerically, the EDOS is obtained from calculating the spectral function with a Lanczos approach 13 and continued fraction expansion as implemented previously 14 . This is extended to the present case.

SM.2 Description of three simplified exciton models (MUE, MUE+EPC, MUE+EC) and the full model
It is convenient to discuss different simplified models for gaining insights. We first introduce the models and discuss the results of the EDOS simulations afterwards.

(M)anifold of (U)ncoupled (E)xcitons -This model (MUE) considers excitons with vanishing coupling.
Vibrations are decoupled. Its Hamiltonian reads where the indices ℎ( ) run over the frontier orbitals of HOMO-type (LUMO-type) centered on the respective molecular sites. This Hamiltonian is diagonal in the indices ℎ and since no transfer integrals are present. The non-zero matrix elements ℎℎ ( ) describe the ionisation energy (electron affinity) at the molecular sites located at ℎ and . The Coulomb matrix elements ℎℎ ( − ℎ ) are given by Supplementary Equation 4 implying a long-range electron-hole interaction following a Coulomb law for the spatial distances (on the fcc lattice) in C60 15 . The strength of the attractive potential for the closest CT-states is given by the material parameter CT that is obtained from ab initio calculations (see section SM.4). The intra-molecular excitations (for = ℎ ) are modeled with an interaction parameter ME of the molecular exciton that amounts to 1.3 eV for C60 as calculated in literature 16 . The corresponding model Hamiltonian reads

MUE+EC
In C60 the transfer integrals are not very large when compared to other small-molecule systems where they can exceed 100 meV 17 . On the other hand, a large number of similar transfer integrals due to the dense packing and the spherical shape of the molecules may enable charges or excitons to delocalise over several molecular units 18,19 . We simulate the EDOS, where the electron-phonon coupling is set to zero, i.e. ( ) = 0, thus, no thermal disorder is present. The rotational degrees of freedom of C60 may still result in a variation in the transfer integrals, which we include by the orientation dependence of the electronic/excitonic transfer integrals between the molecules. Specifically, we choose randomly distributed next-neighbor transfer integrals with zero mean and Gaussian standard deviation TI of 17 meV for all kinds of transfer integrals in the simulations, which are assigned to the next neighbors for each molecule. We find this to be a typical value for the pair coupling of C60 LUMO states when oriented randomly 20 . We apply this value both for donor and acceptor molecules. MUE+EPC -This model extends MUE by including the vibrations and their local EPC. According to section SM.1, coupling to low-frequency modes manifests as a thermal disorder potential ( ). Regarding vibrations with higher frequencies, we only consider the 0-0 transitions as discussed above. The original Hamiltonian of MUE is thus extended according to with ( ) = ℎ ( ) + ( ) where ( ) = ∑ ℏ �(1 + 2〈 〉) † describes the thermal disorder of the LUMO-type orbitals and ℎ ( ) = ∑ ℏ ℎℎ �(1 + 2〈 ℎ 〉) ℎ ℎ ℎ ℎ † describes the thermal disorder of the HOMO-type orbitals. The quantities (or ℎ in full analogy) describe independent Gaussian random variables with standard deviation 1. The temperature dependence of the disorder is given by �(1 + 2〈 〉) with 〈 〉 the mean occupation of mode . We measure the strength of the disorder ( ) by its standard deviation This model results in an EDOS that can be written as The EDOS in Supplementary Equation 19 represents the limit of superimposed localised excitons (see main text) whose linewidth is induced by the intra-molecular vibrations. The shape of each individual transition is a Gaussian function of width loc .
Full model -In the full model that we employ in our simulations, all terms in the above models are included (i.e. the electronic/excitonic couplings and EPC). In this case, the Hamiltonian of MUE+EC is extended according to The so-defined Hamiltonian represents our standard model for the simulation of the EDOS ( ).  . This broadening mechanism can be understood as minimal extension of MUE and may describe a source of disorder, for instance, deviations from a perfect geometry.

Simulation of EDOS and its tails for the different models
As a second mechanism, we study the vibration-induced mechanism by low-frequency vibrations and extend MUE to MUE+EPC. The resulting EDOS (black in Fig. 1b in the main text) shows a strong broadening effect with a CT-exciton band whose tail reaches deep into the gap. This is an unavoidable consequence of the molecular motion at room temperature. The shape of the CT-exciton band for this model can be evaluated analytically and yields a Gaussian-shaped profile with standard deviation loc (see Supplementary Equations 6 and 7 which allows us to validate the numerical results obtained with Equation 3 of the main text. For the numerical ( ), we measure a standard deviation of loc num =70.7 meV by fitting a Gaussian, which agrees with the analytical broadening loc =68.9 meV, thus, further corroborating our numerical approach.
The third mechanism creating exciton bands is the electronic and/or excitonic coupling (MUE+EC) between states. Such couplings are commonly associated with the delocalisation of states and should improve charge transport and increase exciton splitting rates 18 in contrast to the case of disorder or molecular vibrations. This difference between the three mechanisms highlights the ambivalent nature of the width of CT exciton bands and requires the disentanglement of different effects. The resulting lowenergy EDOS (red in Fig. 1c in the main text) exhibits a band around the energy CT 1 which -in absence of vibrations (and hence EPC) -has an effective half band width of CT FWHM/2 ≊ 90 meV. This band is dominated by transfer integrals that are responsible for delocalisation of the electron over several LUMOtype orbitals on C60 molecules. In contrast, the hole wave function at the rubrene donor molecule remains almost localised. The exciton band is therefore a consequence of the high connectivity of the C60 molecules in the first shell around the donor leading to a rich exciton manifold despite random orientation of the molecules. Previous experiments have suggested that electrons can rapidly delocalise in a related set of CT states hosted by C60 clusters 18,19 . Intriguingly, although the observed band width is comparable to the case of EPC-induced bands in Fig. 1b in the main text, the tail at the low-energy side of this band is better described by an exponential energy dependence instead of a Gaussian line shape (see MUE+EPC).
Interestingly, exponential line shapes have been reported in recent experiments on high-efficiency solar cells utilizing non-fullerene acceptors 10 .
After disentangling the individual linewidth contributions, we simulated the EDOS including the intramolecular EPC and the EC simultaneously (see blue line in Fig. 1c in the main text). We measure a total Gaussian broadening of the EDOS tail of tot =69.7 meV that is slightly decreased compared to the value solely induced by the intra-molecular EPC. Although taking into account the 180 meV bandwidth due to the EC, we observe no increase in the CT linewidth. The decrease of the total broadening reflects the correlation of both mechanisms. However, we conclude that thermal disorder dominates the linewidth resulting in the observation of Gaussian tails for those donor-acceptor blends in agreement with experiments.

SM.3 Electron-vibration coupling for various molecular species.
Frozen phonon method for mode resolved EPC -The EPC constants and the mode energies are calculated with the frozen phonon method using the GAUSSIAN09 software package at the B3LYP/6-311G** level of theory [23][24][25][26][27] for the isolated donor molecule in the gas phase. The EPC constants ℎℎ for the HOMO and for the LUMO of C60 are calculated with the NWchem 6.5 software package 28 at the B3LYP/6-311G** level of theory. The mode resolved dimensionless coupling constants ℎℎ (analogously for ) are calculated as (see references 17,29 for details) where is the amplitude of the phonon normal modes in units of Å (amu) 1/2 .

EPC and HOMO/LUMO degeneracy in C60
To describe the vibration-induced bandwidth of donor-acceptor CT-states and of excitonic states in pristine C60, an efficient description (based on effective single HOMO and LUMO states in consistency to the general electronic model) can be worked out with an averaged coupling. For an electronic state equally weighting the 3 LUMO orbitals according to � C 60 , we obtain an effective coupling from an average of the three LUMO couplings of a given molecule. The analogous procedure applies for the HOMOs. Consequently, for a molecular exciton on a single C60 molecule the vibronic coupling results from an equal-weighted superposition of all 15 HOMO-LUMO combinations � C 60 ME � = � C 60 LUMO �� C 60 HOMO � of the product states.
We emphasise that the EPC induced broadening of the individual orbitals is different from the broadening of the molecular exciton. The individual HOMO and LUMO contributions lead to a total broadening of which is equivalent to Supplementary Equation 18. In particular for CT-states we obtain an EPC induced broadening according to Supplementary Equation 24 for both donor-C60 and C60-C60 CT-states using the effective EPC constants eff and ℎℎ eff for the C60 orbitals. In contrast, for the molecular exciton in C60 we This differs from Supplementary Equation 24 and might lead to an enhanced or reduced broadening depending on the relative sign of the coupling constants eff and ℎℎ eff i.e. the broadening might be partly cancelled if eff = − ℎℎ eff for a certain mode .

SM.4 Ab initio calculations of polarisation and CT-energies for donor:C60 blends
Equations for the CT-energy, exciton binding energy, and polarisation corrections -The aim of this section is to obtain the electron-hole interaction energy CT and the charge-transfer energy CT of a donor-acceptor pair in the organic film and to describe the approach to their simulation. We consider that this pair is diluted in the host material, which is taken to be C60 fullerenes. In general, the theoretical CT- We focus first on the simulation of CT for a relaxed geometry of a donor-acceptor dimer. The simulations are performed in gas phase and polarisation corrections are subsequently added to account for the environmental screening Here Δ 0 is the electron-hole interaction energy in the gas phase, which is calculated for a fixed dimer geometry as We transform this equation slightly to with the CT energy for the dimer in vacuum The involved energies (… , … ) in Supplementary Equations 28 to 30 represent the DFT total energies of the dimer for a certain charging state of the donor (D) and the acceptor C60 (A) as indicated by superscripts to the species. The correct charging is ensured by using constrained DFT 31 within the NWchem 6.5 package. The energies were calculated with the CAM-B3LYP/6-311G** level of theory 32,33 and the additional inclusion of Grimme's empirical dispersion correction GD3 34 and asymptotic correction LB94 35 .
To obtain CT , we simulate the polarisation correction to Δ 0 namely CT . According to the three energy terms in Supplementary Equation 29, CT has three contributions: where the parentheses indicate the molecules whose charging state is changed. For instance, the energy Combining Supplementary Equations 26, 27, 31 to 33, we find the CT-energy as In subsequent sections, we discuss the approaches to the individual terms in Supplementary Equation 34.

Gas-phase energies
where is the dielectric constant (relative permittivity) of the continuum either composed of donor or acceptor molecules. We assume the continuum being outside a sphere with radius that contains the same volume as the respective molecule. The volume is calculated based on the geometrical extend of the molecule according to its principal semi-axes , , . Thus is calculated as the geometric mean of the principal semi-axis , , as = ( )

Bulk polarisation corrections for dimers -The dipole polarisation correction A dimer (DA) in
Supplementary Equation 31 of the excited dimer composed of a negatively charged C60 and a positively charged donor in the C60 environment is calculated within the continuum approach 16 as is the dielectric constant of C60 which is set to 4.4 37 . is the electric dipole moment of the dimer, which is calculated from the respective Löwdin-charges during the NWchem single-point calculation. Finally, is the radius of a sphere that contains the same volume as the dimer. The dimer volume is calculated based on its principal semi-axes , , as the volume of the corresponding ellipsoid.
The polarisation correction of the monopole charges in the dimer geometry is given by the quantities In Supplementary Table 3 we summarise the obtained material parameters taken for the simulation of the EDOS ( ).

SM.5 Analytical model for maximum open-circuit voltage
Open-circuit voltage -The open-circuit voltage for a solar cell is given by 48,49 Here SC is the charge current at short-circuit conditions and 0 is the dark saturation current.

Shockley-Queisser-model for � and extension to a broadened absorption tail -To quantify the radiative
losses of the open-circuit voltage r due to a finite absorption tail, we study the following analytical model.
Firstly, we calculate numerically the radiative limit of the open-circuit voltage in the Shockley-Queisser (SQ) model with a gap energy g defined as the absorption maximum of the lowest CT-peak, i.e. g = CT,peak = CT 1 . In this limit the photovoltaic external quantum efficiency ( ) is given by a step function with an infinitely steep absorption edge at g In the radiative limit, the open-circuit voltage of the SQ-model becomes with the short-circuit current from the absorbed solar photons (43) and the dark saturation current due to the emitted thermal photons according to Secondly, in presence of a broadening mechanism (e.g. induced by the EPC of intra-molecular modes) that smears out the absorption edge, the open-circuit voltage is reduced below the limiting r SQ . To study this effect for our results for the temperature dependent broadening ( ) of the EDOS, we extend the SQmodel for the EQE by adding a tail according to The short circuit and the dark saturation currents then become Finally, the radiative open-circuit voltage takes the form r SQ+t = r SQ − Δ r (50) where the deviation from the SQ-limit can be expressed as Here, a voltage gain that arises due to the presence of a finite absorption tail (second term on the rhs. of  Fig. 3c, 3d, 4c and Fig. 5a of the main text. We therefore find for em,0

Analytical expression for � and quantification of the losses
which is dominated by the leading term For the dark saturation current of the Gaussian tail we have where the energy integral is solved explicitly with the result where (57) The voltage loss Δ r depends on the linewidth 2 ( ) as follows (45)(58) We now consider the limit of small and large broadening ( ) leading to qualitatively different results for the voltage losses Δ r . For large broadening i.e. B ≪ ( ) ≤ g we find whereas for small broadening i.e. B ≤ ( ) ≪ g or ( ) ≤ B ≪ g , we find

SM.6 Systems with non-fullerene acceptors
EDOS simulations in non-fullerene systems -In the next paragraphs we discuss the material parameters and the subsequent EDOS simulations involving the non-fullerene acceptor materials boron subnaphtalocyanine chloride (SubNc), boron subphthalocyanine chloride (SubPc), and chlorinated SubPc (Cl6-SubPc).

Energy levels in SubNc, SubPc and Cl6-SubPc -For the non-fullerene acceptor materials SubNc, SubPc and
Cl6-SubPc we use reference values for the ionisation potential and the electron affinity from the respective peak positions of UPS and IPES measurements 52 Fig. 5b in the main text) yielding a binding energy of ME =1.17 eV. The absorption maximum of Cl6-SubPc is slightly lower due to chlorination as compared to SubPc 54 at 2.05 eV (see Table S24 of the supporting material in the according reference). The binding energy of the molecular exciton in Cl6-SubPc then is given by ME =1.2 eV.

Material parameters for the EPC and EC in SubNc, SubPc and Cl6-SubPc -We calculated the EPC in SubNc,
SubPc and Cl6-SubPc using the method described in section SM.3.
Note that the monomer orbitals |Φ ⟩ calculated in the DFT-simulations are in general not orthogonal.
Hence a Löwdin orthogonalisation | ⟩ = − 1 2 |Φ ⟩ is performed to orthogonalise them where is the Hamiltonian of the dimer (Fock matrix) and the overlap matrix of the non-orthogonal monomer orbitals ′ = ⟨Φ |Φ ′ ⟩. The Hamiltonian , the overlap matrix and the monomer orbitals |Φ ⟩ were obtained by DFT-calculations with the Gaussian16 package 58 and the B3LYP level of theory combined with the 6-311G** basis set. Moreover, we find that SubNc is not an exception for this compensation effect. In Supplementary Table 5 we briefly summarise the calculated values for the EPC-induced broadening of the individual molecular orbitals with that of molecular excitons for SubNc, SubPc and Cl6-SubPc. All obtained values support our design strategy that the EPC-induced broadening may be significantly reduced for molecular excitons as

SM.7 Homogeneous broadening of an excitonic state in the single-state limit
Homogeneous broadening of an excitonic-state and of the zero-phonon line -The broadening of lowenergy tails due to low-energy vibrations that we study in organic solar cell films is derived from molecular vibrations and is additionally affected by the electronic coupling (EC) to the surrounding host. If the surrounding is homogeneous, the broadness of the tails is everywhere the same in the sample, hence the denotation homogeneous. A different quantity (although also of homogeneous type) is the molecular homogeneous linewidth of the zero-phonon line (ZPL) that can be measured for single molecules close to zero Kelvin 59 . The difference is explained in detail here below.
Analytical result for the homogeneous broadening in the single-state limit -To clarify the difference, we consider the limit of vanishing electronic coupling to the surrounding, i.e., an isolated excitonic state in the low-energy absorption spectrum. We will demonstrate that even for very low temperatures (here for 4 K) we find an effective homogeneous broadening for excitations of that state that is induced by the EPC and which can be of several tens of meV. This is in contrast to the homogeneous broadening of the ZPL that can be below 1 GHz for some molecules.
In the single-state limit, the single-particle spectral function ′( ) (density of states), according to Feynman's disentangling theorem 60 for operators, is given by the well-known form 61 with the electronic onsite energy of the LUMO (for the HOMO in full analogy) and the reorganisation energy . The time dependent correlation function Φ ( ) simplifies in the present case of dispersionless vibrational modes with circular frequency (mode index ) and intra-molecular coupling constants with which is a series of discrete peaks for each superposition of the energies ∑ ℏ weighted with the temperature dependent factors that include the modified Bessel functions �2� � 2 �⟨ ⟩(1 + ⟨ ⟩)� that are evaluated at the given temperature with the thermal occupation ⟨ ⟩ of mode . The product of these functions leads to the spectral function in the single-state limit. This is an exact result at vanishing electronic coupling.
Analogously, for the isolated excitonic state that we consider as a model, we find the excitonic twoparticle spectral function (EDOS) with the energy of the state CT = − + ℎℎ − ℎℎ − ℎ ℎ and the correlation function Φ ℎ ( ) describes the phase dynamics of a LUMO electron at site and a HOMO hole at site ℎ. The analytical expression for the spectral function in the single-state limit is the convolution of contributions from all intra-molecular modes of the associated donor (D) and acceptor (A) molecules according to indicated by the mode index that, in the case of a CT state, runs over donor and acceptor molecular modes.
In Supplementary Figure 1 Supplementary Figure 1 shows that the lower the energy of the strongly electron-phonon coupled modes the closer are the discrete peaks. We observe that a continuous situation can easily occur for a moderate number of low-frequency modes.
Supplementary Figure 1 illustrates the different linewidths under discussion. The multiple vibrational sublines of the considered isolated transition create an effective broadening. In dependence on the density of the electron-phonon coupled low-energy vibrations, it can be an effective broadening of a continuous distribution. We note that the transition between a dense set of discrete peaks (rubrene:C60) and a continuous distribution is triggered by an increasing number of low-frequency modes. In addition, a more continuous distribution is observed with finite temperatures above 80 K or even at room temperature with a broadness that is the broadness loc ( ) in Supplementary Equation 18 induced by the thermal disorder potential ( ). In addition, the transition to a continuous distribution is much more likely if (beyond the single state model considered here) additional transfer integrals are considered as in the organic thin films studied in the main text. Because of these transfer integrals in the thin film, one cannot observe any discreteness in the spectra for none of the systems. This effective broadening is the relevant quantity for absorption in organic solar cells since the entire broadened electronic transition absorbs the light. Hence, we call it the broadening due to zero-point vibrations in this work.
From the standpoint of single-molecular spectroscopy, this might be termed differently (e.g. 'effective broadening') because it is to be distinguished from the homogeneous broadening of the zero-phonon line (ZPL) which describes the broadness of an individual peak associated to a zero-phonon (0-0) transition between the vibrational ground states of all modes. The latter can be several orders of magnitude smaller (GHz or even tens of MHz). 59 From the spectra in Supplementary Figure 1 one can understand that only for specific molecules the broadening of the ZPL can be observed, namely for those with a very sparse vibration spectrum of strong coupling modes. Even for the TAPC:C60 case it is not visible. In contrast, for the molecules used here the denser vibrational spectrum creates a more continuous absorption spectrum (see above paragraph).
We emphasize that even in the case of the considered isolated transitions and ultralow temperatures it is generally very difficult to observe a ZPL and its linewidth, which was achieved recently 59 . In the context of the present work, this is not relevant because it is many orders of magnitude smaller. Instead, we need to consider the low-frequency-modes-induced effective linewidth seen in Supplementary Figure 1 because we are concerned with the absorption properties of the organic film, which is relevant for the radiative voltage loss.

SM.8 Molecular insights into EPC and tuning thereof
Generally speaking, the electron-phonon coupling is strongly connected to the structure of the considered molecular orbital and its impact on the molecular structure. For instance, when charging a molecule negatively, the deformation of the molecule from its neutral relaxed state to its anionic relaxed state is dominantly affected by the singly occupied LUMO. At bonding lobes of the orbital, bond lengths are shortened, whereas at anti-bonding nodes, bond lengths are increased. These deformations can be described by combining vibrational normal modes with the EPC values characterizing the energetic contribution to the relaxation of the individual normal modes.
These basic considerations show the connection between the molecular structure and the EPC values and led to design strategies proposed in several systematic theoretical studies. In general, organic molecules can be roughly classified into three groups: (i) molecules with dominant high-frequency modes, (ii) molecules with dominant low-frequency modes, and (iii) molecules with a balanced contribution of normal modes over the entire frequency range 9 . Class (i) includes mostly relatively compact molecules with the orbital being delocalised over a π-conjugated core structure, such as rubrene. This behaviour applies to the entire class of oligoacenes, where the EPC values decrease with increasing number of phenyl rings 29 . In fact, orbital delocalisation is an established design strategy for reduced EPC values over the entire frequency range as the deformation of the molecules upon charging is reduced and the effect of a single molecular orbital on the molecular geometry is weakened 62 . Such a manipulation of the molecular structure also leads to smaller contributions by zero-point vibrations.
However, an extended π-conjugated core structure does not necessarily prevent from contributions by low-frequency modes as demonstrated for more discotic molecules such as triphenylene, perylene, or hexabenzocoronene 29,63,64 . The transition from group (i) to group (ii) is gradual. An increased flexibility of the molecular core leads to increased EPC values. This is also observed when comparing pentacene with thiophene-based analogues 65 .
The effect of flexibility is even more pronounced when attaching side chains to the molecular core. When comparing the EPC values of tetracene with the ones of rubrene, one observes noticeable contributions by low-frequency modes by side-group bending vibrations, although the frontier molecular orbitals are widely restricted to the tetracene core 66 . This is even more pronounced for molecules with distinct orbital delocalisation over flexible side groups as demonstrated for the material class of aza-BODIPYs 62,67,68 .
Increased orbital delocalisation can decrease the EPC values, however, the side groups especially induce additional contributions by low-frequency vibrations. It has been demonstrated for this material class that the design strategy of rigidification can lead to a significant reduction of EPC values. There are multiple avenues for reducing the flexibility of the side groups including the use of intramolecular hydrogen bonds, steric hindrances, or covalent bridges.
Molecules of group (ii) with dominant EPC values by low-frequency modes often exhibit a torsionally flexible molecular core with delocalised molecular orbitals, such as m-MTDATA or oligothiophenes 69 . Also for these molecules, delocalisation of the orbitals can effectively reduce the EPC values over the entire