Optical Properties of H-Bonded Heterotriangulene Supramolecular Polymers: Charge-Transfer Excitations Matter

H-bonded N-heterotriangulene (NHT) supramolecular polymers offer a nice playground to explore the nature and dynamics of electronic excitations in low-dimensional organic nanostructures. Here, we report on a comprehensive molecular modeling of the excited-state electronic structure and optical properties of model NHT stacks, highlighting the important role of intermolecular charge-transfer (CT) excitations in shaping their optical absorption and emission lineshapes. Most importantly, we show that the coupling between the local and CT excitations, modulated by the electric fields induced by the presence of polar amide groups forming H-bonded arrays along the stacks, significantly increases the resulting hybrid exciton bandwidth. We discuss these findings in the context of the efficient transport of singlet excitons over the μm length scale reported experimentally on individual self-assembled nanofibers with molecular-scale diameter.


S1 Tunning of the long-range corrected density functional
Long-range corrected (LC) functionals split the exchange term in a short-range and a long-range component.The first component is represented by the exchange expression of the own density functional whereas the second is evaluated with an "exact" Hartree-Fock exchange expression. 1 LC functionals significantly depend on the ω parameter, which measures the threshold distance for the short-range and long-range region.Optimization of the ω parameter for an LC functional applied to a donor−acceptor (D−A) supramolecular complex is highly recommended to obtain acceptable energies for the CT excited states and satisfy the Mulliken rule E CT = IP + EA -1/r, 1 where IP, EA, and r correspond to the ionization potential, the electron affinity, and the distance between the donor and acceptor units, respectively.Here, we have performed an optimization of ω by minimizing the J(ω) function (Eq.S1) that provides the deviation of the HOMO energy ( , w e S HOMO ) of the neutral system with respect to the IP plus the deviation of the HOMO energy of the anion system with respect to the EA.

S2 Diabatization based on the fragment particle-hole densities (FPHD)
The diabatization scheme 4 based on the fragment particle-hole densities is a method developed to include simultaneously local excited states (Frenkel states, FE) and chargetransfer (CT) states, which is a non-trivial task.For instance, in the simple case of a dimer, four type of diabatic states can be found: two FE states and two CT-states.By using the hole and electron population of each fragment, the FPHD diabatization allows to set a property that univocally describes both the local and the CT states.That means that 2N matrices (holes and electrons) are simultaneously used, where N is the number of fragments.Figure S2 displays the 4 diabatic property matrices associated to the diabatic states.As in the adiabatic representation the four matrices collecting the hole and electron transition density populations are non-diagonal, the FPHD method requires of a unique unitary transformation matrix (C) that performs the adiabatic-to-diabatic transformation to the 2N property matrices simultaneously.
The diabatic states are those that have a maximum localization of the hole and electron populations.This maximum localization is achieved through simultaneous diagonalization of the 2N matrices, what is accomplished by the Jacobi sweep algorithm. 5ce the adiabatic matrices are diagonalized, the diabatic states can be classified according to their nature, and each subspace of a specific nature must be diagonalized to ensure that the couplings between states of the same nature are zero.This procedure thus implies the diabatization between local FE states or CT states of equivalent nature, and therefore provides simultaneously electronic and excitonic couplings.

S3 Optical and electronic properties of aggregates
The Frenkel-CT Holstein (FCTH) Hamiltonian used to simulate the optical and electronic properties of the NHT-based molecular aggregates can be written as: Where the Frenkel part in the nearest neighbor's approximation reads as: where E represents the energy of a Frenkel exciton in the site n in the aggregate, J is the excitonic coupling, ω 0 the frequency of the effective normal mode that modulates the energy of the excited site and 2 * l is the Huang-Rhys factor of the Frenkel state (S 1 ) that measures the relative shift with respect to the S 0 .Similarly, we define a CT Hamiltonian (Eq.S4), which is diagonal since we assume only interaction between consecutive sites, as:

S6
The full Hamiltonian is expressed in a delocalized basis set including one-and twoparticle states for FE excitons and two-particle states for CT states.Therefore, the a th eigenstate wave vector is expanded as: are the corresponding two-particle CT states.
In this contribution, we use a basis cutoff with the maximum number of quanta   is limited to 5. The intensity of the absorption spectra is computed as: where 0 , 0 a k m = is the transition dipole moment between the vibrational ground state and eigenstate a at the high symmetry k point, and the exponential part adds a homogeneous broadening with a standard deviation (σ) of 0.05 eV.The transition dipole moments are estimated by rotating the diabatic transition dipole moments with the eigenvector matrix from diagonalizing the model Hamiltonian from Eqs. S2-S6.In a similar way, the emission intensity is computed as: where the summation over all the receiving states, denoted with symbol w, is incorporated.The receiving states are the one-and two-particle vibronic basis of the electronic ground state.Thus, , 0 wa k m = is the transition dipole moment from eigenstate a to the w vibronic ground state with a total number of v vibrational quanta.The emission intensities are thermalized by weighting the intensities according to a Boltzmann distribution (Eq.S10) on the emissive states.
To include static disorder in the simulation of the optical properties, we compute the absorption and emission spectra of 1000 replicas where the values of E, E +-, and E -+ are randomized independently according to a Gaussian distribution with the mean centered in the reference values (see Table S4 below) and a standard deviation of 130 meV in a non-correlated manner.Therefore, all local (E) and CT (E +-and E -+ ) excitations are likely to be different at each realization and in between realizations.Finally, we obtain the average spectra of the 1000 realizations.
Radiative lifetimes (t ) are estimated as an average of the lifetime computed for each realization ( , Eq. S11), and the radiative decay rate k rad,i is computed as a summation over the thermally weighted emission line spectrum (Eq.S12).
In Eq.S12, ω a,k=0 is the energy of the emissive state, 0 e the vacuum dielectric permittivity, h the reduced Planck constant, and c the speed of light.Note that Eq.S12 requires that ω a,k=0 and ω 0 should be in the same energy units.

S4 Electronic structure of monomer and dimer systems
The electronic structure and the excited states of a N-centered heterotriangulene (NHT) symmetrically substituted with three acetamide groups (Figure S3a) were studied at the B3LYP-D3/6-31G** level after its previous geometry optimization within C 3 symmetry restrictions.Figure S3b displays the computed frontier molecular orbitals leading to a single HOMO energy level and a double-degenerated LUMO belonging to the irreducible E symmetry representation.Excited states were computed using the Tamm-Dancoff approximation (TDA-DFT), 6 which predicts the two lowest-energy singlet electronic transitions (S 0 → S 1 and S 0 → S 2 ) degenerate in energy at 2.95 eV (Table S1) and mainly described by single HOMO → LUMO 1 and HOMO  LUMO 2 monoexcitations.The  S4).The low-lying singlet excited states of the dimer models were computed within the TDA-DFT approach using both the B3LYP-D3 and OT-ωB97XD functional.For the dimers, hereafter denoted as AB, both density functionals provide a similar picture (Table S2), where the lowest singlet excited states (S 1 and S 2 ) exhibit a remarkable CT character (A -B + states), and are followed at higher energies by two pairs of FE-type A * B and AB * excited states (S 3 /S 4 and S 9 /S 10 ) and two CT A + B -states (S 13 /S 14 ).A more detailed analysis of the data reported in Table S2 reveals some significant differences It is noteworthy that the H-bonded supramolecular arrangement in the C 3 -symmetry dimer causes an energy splitting between the two pairs of CT excitations (A + B -and A - B + ).This lifting of degeneracy is due to the electrostatic stabilization (destabilization) of the CT excitations that are polarized parallel (antiparallel) to the electric field generated by the amide groups.As a matter of fact, excited-state degeneracy is recovered when either increasing the intermolecular distance d between the planes of the two molecules constituting the H-bonded dimer (Figure S5) and, particularly, when forcing the amide groups to remain in the molecular plane, thus effectively breaking the triple H-bond array (Figure S6). Figure S5 clearly reveals that the CT states (A + B -and A -B + ) shift to higher energies when increasing the intermolecular distance (in line with Mulliken's law).
Simultaneously, a reduction of the energy splitting between the A + B -and A -B + states takes place owing to the smaller interaction of the holes and electrons with the permanent dipole moment of the other molecule.Although the analysis of the excited states is more difficult for the purely π-stacked dimer with no H-bonds due to the delocalized nature of the molecular orbitals involved in the excitations, Figure S6

S5 Diabatic model Hamiltonian
Table S3 displays the FE and CT excitation energies (ΔE FE and ΔE CT ), the excitonic couplings (J) and the hole and electron transfer integrals (t h and t e ) computed for the three different dimers with decreasing dipole moment along the stacking direction: H-bonded C 3 , H-bonded C 1 , and non-H-bonded C 3 dimers.The energy parameters given in Table S3 were estimated within the FPHD diabatization protocol using the results obtained for the excited states of the dimers at the TDA-DFT OT-ωB97XD/6-31G** level.On one hand, the FE-type diabatic states are found for the three dimer models in a narrow energy window between 3.40 and 3.55 eV.However, as anticipated from the analysis of the adiabatic states, the energetic position of the CT states changes much more along the different dimer models.In particular, the CT states are centered around 3.55 eV for the non-H-bonded C 3 -symmetry apolar dimer.When a dipole moment due to the amide groups orientation is introduced as a consequence of H-bond formation, an energy splitting between the A + B -and A -B + diabatic states appears, which is significantly higher for the H-bonded C 3 dimer (1.12 eV) than for the H-bonded C 1 dimer (maximum 0.62 eV) due to the higher dipole moment of the former.On the other hand, the values calculated for the couplings J, t h , and t e are in the same range without being affected by the increase or decrease of the dipole moment.Therefore, to check specifically the influence of the E FE-CT energy offset, the couplings obtained for the C 3 -symmetry dimer are used in the following simulations (Table S4).
Table S3.Diabatic energies and couplings obtained at the OT-ωB97XD level by using the FPHD diabatization scheme for three different dimer models.x and y subscripts are used to describe the main orientation of the transition dipole moment to distinguish between degenerate states.A schematic picture of a C 3 -symmetry dimer, with molecules A and B and the direction of the local (red arrows) and total (black arrow) dipole moments Another important aspect of our model is that the states of the same nature degenerate in energy are condensed in a single effective state for the modelling of the optical and excitonic properties.A single effective state model is generally used to simplify a complex situation when multiple states degenerate in energy are present (as it is the case here for the NHT-based system).The main idea is based on establishing an effective coupling between two states that would have the same electron/exciton transfer rate constant as a global rate constant involving multiple electron/exciton transfer processes (from one state at one site to different states degenerate in energy at another site).10] Table S4 contains the effective couplings that have been finally used in the model together with the energies for the different states (E, E +− and E −+ ).The sign of the effective coupling is set to coincide with the sign of the highest direct coupling (Table S4).Regarding the transition dipole moments for the FE states ( n ), we employed the transition dipole moment of the isolated molecule for the excited state S1 (Table S1) for the first FE electronic excitation (site 1).For the rest of the FE excitations, the transition dipole moment was rotated along the z axis according to the (n−1)•36º, where n corresponds to the site number in Eq.S3.The CT states were assumed to be completely dark and, therefore, the transition dipole moments are set to be 0.
To compute the optical properties of the aggregates (k = 0), we need 10 molecular sites to get a full helical pitch.This gives rise to 30 electronic states, 10 Frenkel states and 20 nearest neighbor CTs including those necessary due to the periodic boundary conditions.
Likewise, we use a basis cutoff with the maximum number of vibrational excited quanta

Figure 3 Figure S1 .
FigureS1provides the tuning of the ω parameter for the NHT molecule using the LC

Figure S2 .
Figure S2.a) Diabatic matrix representation of the hole (h) and electron (e) transition density populations of each fragment (A or B) for the four diabatic states.Labels indicating the diabatic states are provided at the right side.b) Hole (red) and particle (blue) transition densities are represented for each state, using an anthracene dimer as example model.

† n b and n b are the
creation and annihilation operators of a vibrational quantum in the unshifted S 0 nuclear potential well on-site n.
v are the local one-and two-particle FE states, where the exciton is at site n, and and n m v w are the vibrational quanta in sites n and m, respectively, and

Figure S4 .
Figure S4.Representation of the three ideal dimers built based on models I and II and use for further parameterization of the model Hamiltonians: a) H-bonded C 3 -symmetry dimer, b) H-bonded C 1 -symmetry dimer and c) non-H-bonded C 3 -symmetry dimer.
shows a set of CT-type excited states whose energy increases with the intermolecular distance and are now degenerate in energy.These findings highlight how the orientation of the amide groups tune the permanent dipole moment within the supramolecular arrangement and, consequently, modulate the relevant E FE-CT energy gap.The three dimer models (FigureS4) are used to delve into how the energy position of CT states influences the optical and excitonic properties of the NHT-based supramolecular polymer (vide supra).

Figure S5 .
Figure S5.Energy diagram of the low-lying FE and CT states of the C 3 -symmetry Hbonded NHT dimer at different intermolecular distances (d).A representation of the dimer

Figure S6 .
Figure S6.Energy diagram of the low-lying FE and CT states of the C 3 -symmetry non-
3 -symmetry) was constructed where the amide groups are kept in the molecular plane of the π-conjugated core being not able to form intermolecular H-bonds (Figure

Table S2 .
Vertical excitation energy (∆E) calculated for selected adiabatic singlet excited states of the ideal NHT dimers at the DFT level using the B3LYP-D3 and OT-ωB97XD (ω = 0.15 Bohr - 1 ) functionals.The selected adiabatic excited states are those that can be easily associated with quasi-diabatic states of FE-type (A * B and AB * ) and CT-type (A + B -and A -B + ) nature.

Table S4 .
Set of parameters used for the subsequent modelling of the optical and excitonic bands in the different situations.S , S + , and S -) obtaining values of: 0.481, 1.751, and 0.549 for * S , S + , and S -, respectively.→ S 1 electronic transition (2.95 eV) and the relaxation energy of the S 1 excited state (0.072 eV).The proposed model provides a good agreement with respect to the experimental spectra, especially for the relative intensity of the peaks defining the vibrational progression. 11The model used to calculate the spectra of the NHT monomer shows no Stokes shift since solvent effects, which are responsible for Stokes shifts in single rigid molecules, have not been included.Nevertheless, the model is able to account for the Stokes shift resulting from the supramolecular aggregation in the dimer, which is more relevant in this context. *