Nature of Optical Excitations in Porphyrin Crystals: A Joint Experimental and Theoretical Study

The nature of optical excitations and the spatial extent of excitons in organic semiconductors, both of which determine exciton diffusion and carrier mobilities, are key factors for the proper understanding and tuning of material performances. Using a combined experimental and theoretical approach, we investigate the excitonic properties of meso-tetraphenyl porphyrin-Zn(II) crystals. We find that several bands contribute to the optical absorption spectra, beyond the four main ones considered here as the analogue to the four frontier molecular orbitals of the Gouterman model commonly adopted for the isolated molecule. By using many-body perturbation theory in the GW and Bethe–Salpeter equation approach, we interpret the experimental large optical anisotropy as being due to the interplay between long- and short-range intermolecular interactions. In addition, both localized and delocalized excitons in the π-stacking direction are demonstrated to determine the optical response, in agreement with recent experimental observations reported for organic crystals with similar molecular packing.


Experimental methods
Atomic force microscopy (AFM) images are collected in intermittent-contact mode in air with a Nanoscope V MultiMode (Bruker) using silicon probes (force constant 40 N/m, resonance frequency 300 kHz, tip radius 8 nm) with a resolution of 512 × 512 pixels.
Polarized optical absorption measurements are performed at normal incidence in the spectral range from 1.7 to 3.6 eV by using a Perkin-Elmer Lambda900 spectrometer, equipped with a depolarizer and Glan-Taylor calcite polarizers, with a light spot size of about 5 mm 2 .
Plots of the molecular and crystal structure were produced with Mercury CSD 3.7. 1

Experimental configuration in optical measurements
The absorption spectra reported in Figure 1e in the main text are collected on a 1 nm thick ZnTPP film. These spectra are a good approximation of those of a (100)-oriented ZnTPP single crystal when the light beam is normal to (100) ZnTPP surface (thus, light propagates along the a * ZnTPP axis) and is linearly polarized with the electric field E parallel and orthogonal to [001] ZnTPP (see Figure S1), notwithstanding (100)-and (100)-oriented ZnTPP crystals could coexist on KAP surface. The reason can be understood looking at Figure S2, where a view along the a * ZnTPP axis for (100)-and (100)-oriented is reported together with the directions of E of polarized light used for collecting spectra. The X (X ) and Y (Y ) directions in (100)-[(100)-] oriented ZnTPP crystals reported in Figure S2 are the orthogonal projections onto the molecular plane of the directions connecting the opposite nitrogen atoms along which the optical spectra of the isolated molecule with the conformation assumed in the crystal have been computed (see Figure 3c in the main text). At normal incidence, the projections of X and Y directions on the sample surface are connected to X and Y by means of a C 2 axis along c ZnTPP axis. It is therefore obvious that at normal incidence no differences could be observed between the optical response of a (100)-oriented ZnTPP single crystal or a couple of crystals oriented like those reported in Figure S2 (both present in our S2 samples), when incident light is polarized along and perpendicularly to the c ZnTPP axis. Note that the above approximation fails under any other configurations (e.g. different directions of light polarization at normal incidence or oblique incidence measurements). Figure S1: Sketch of the experimental configuration of the optical measurements at normal incidence. View of the crystal (one layer) along a * ZnTPP , which is also the direction of light propagation. The directions of light polarization are also reported. Figure S2: View along a * ZnTPP for (100)-and (100)-oriented ZnTPP crystals. For simplicity, just one unit cell is reported. X and Y axes (the same also reported in Figure 1 in the main text) are those used for computing the optical spectra of the isolated molecule with the conformation assumed in the crystal.

Theoretical Methods
The quantum-ESPRESSO package 2,3 is used for DFT simulations while all the MBPT calculations are done using Yambo. 4 We refer the reader to two previous papers on metal-free meso-tetraphenyl porphyrin (H 2 TPP) where a similar theoretical/computational approach and numerical implementation have been used. 5,6 The DFT calculations are based on plane-wave expansion and norm-conserving pseudopotentials and use the generalized gradient approximation (PBE). 7 A semi-empirical van der Waals correction 8 is added in the exchange-correlation term in order to obtain the re- This allows us to obtain not only the occupied but also a high number of Kohn-Sham empty states needed for the excited-state simulations (up to 15 eV above the highest occupied state). To provide a good estimation of the electronic levels (bandstructure), the second step consists in carrying out GW calculations within a perturbative scheme and using the plasmon-pole approximation for the inverse dielectric matrix. 9 For the isolated molecule, due to the high computational cost, we limit the calculation to one-shot G 0 W 0 approach. For the ZnTPP crystal that is the main phase of interest here, we instead apply a more expensive self-consistent GW approach by updating the energies in both G and W . This approach has been shown to improve the agreement of the energetic position of the electronic states with respect to experiment in other organic compounds, as well. 10

S4
The quasi-particle energies are obtained as: where the upper left (lower right) is called the resonant (antiresonant) block, is Hermitian and is defined as where K = W −2v is the excitonic kernel, with W andv being the screened and bare Coulomb interaction without the long-range part, and the factor 2 comes from the spin-degeneracy. 12 The coupling part H coupl = eh|K|h e is symmetric and describes the interaction between the resonant and antiresonant blocks, or in other words, between the eh pairs at positive and S5 negative (antipairs) energies. Here, electron-hole antipairs are denoted by h e while E QP h , |h (E QP e , |e ) refer to quasi-particle energies and eigenfunctions of occupied (unoccupied) states respectively. Starting from the excitonic Hamiltonian H exc , it can be shown 12 that the photo-absorption cross section is proportional to where |0 is the ground-state wavefunction, ξ is the unit vector in the direction of the electric field of the polarized light and D is the electronic dipole, while |P = ξ · D|0 . Using the spectral representation, the cross-section can be rewritten in terms of excitonic eigenvalues E exc λ and eigenvectors A eh λ (which are obtained diagonalizing the excitonic matrix H exc ) as: where |λ = eh A eh λ |eh is the excitonic state expressed as linear combination of independent quasi-electron and holes states. It is then clear that the optical matrix elements, when local-fields and excitonic effects are taken into account, are due to a mixing of independent quasi-particle transitions mediated by the excitonic eigenvectors A eh λ .
BSE spectra: convergence, excitonic oscillator strengths, and Tamm- 3) is truncated. Figure S3 illustrates the effect on the overall anisotropy: the spacing between S6 the peaks increases to 0.43 eV compared with 0.2 eV reported in Figure 3a at full convergence, and the lineshape and relative intensities are distinctly modified.  It is worth pointing out that the Tamm-Dancoff approximation (TDA), which consists in assuming that the coupling blocks are equal to zero, is generally a very good approximation for extended or periodic materials. Nevertheless, since this term is actually dominated by the unscreened Coulomb potential v, which is a measure of the inhomogeneity of the electronic density, the TDA fails for systems where the electronic density is strongly inhomogeneous,

Dancoff approximation
i.e. nanostructures, molecules and molecular crystals where van der Waals interactions are mainly responsible for intermolecular bonds.
S7 Figure S4: Absorption spectra of crystalline ZnTPP showing oscillator strengths of all excitons. Figure S5: Exciton amplitudes A λ (ω) corresponding to the Q, B and B 1 excitons.
As discussed in Ref. 14 the failure of the TDA can be quantified and understood by looking S8 at the amplitude functions defined as Indeed if contributions to a given exciton λ occur also from eh antipairs, peaks at negative energies appear.
We look then at the amplitudes A λ (ω) of the three main excitons Q, B, B 1 , which are reported in Figure S5, where a contribution at negative energies (he antipair transitions) is clearly visible. This explains clearly the need to include the coupling parts in the excitonic Hamiltonian.